PHI LOS 0PM V
OF THE
INDUCTIVE SCIENCES!,
FOUNDED UPON THEIR HISTORY.
BY WILLIAM WHEWELL, D.D.,
MASTER OF TK1\1TY COLLEGE.
A-- NEW EDITION,
WITH CORRECTIONS AND ADDITIONS, AND
AN APPENDIX, CONTAINING
PHILOSOPHICAL ESSAYS PREVIOUSLY PUBLISHED.
IN TWO VOLUMES.
VOLUME THE FIRST.
LONDON:
JOHN W. PARTNER, WEST STRAND.
M.DCCC.XLVII.
REV. ADAM SEDGWICK, M.A.,
SENIOR FELLOW OF TRINITY COLI.IOE,
WOODWARDIAN PROFESSOR OF GEOLOGY IN THE UNIVERSITY OF
CAMBRIDGE, AND PREBENDARY OF NORWICH.
MY DEAR SEDGWICK,
WHEN I showed you the last sheet of my History of the In
ductive Sciences in its transit through the press, you told me that
I ought to add a paragraph or two at the end, by way of Moral
to the story ; and I replied that the Moral would be as long as
the story itself. The present work, the Moral which you then
desired, I have, with some effort, reduced within a somewhat
smaller compass than I then spoke of; and I cannot dedicate it
to any one with so much pleasure as to you.
It has always been my wish that, as far and as long as men
might know anything of me by my writings, they should hear of me
along with the friends with whom I have lived, whom I have loved,
and by whose conversation I have been animated to hope that I
too might add something to the literature of our country. There
is no one whose name has, on such grounds, a better claim than
yours to stand in the front of a work, which has been the subject
of my labours for no small portion of our long period of friend
ship. But there is another reason which gives a peculiar pro
priety to this dedication of my Philosophy to you. I have littlo
doubt that if your life had not been absorbed in struggling
with many of the most difficult problems of a difficult science,
you would have been my fellow-labourer or master in the work
which I have here undertaken. The same spirit which dictated
your vigorous protest against some of the errours which I also
attempt to expose, would have led you, if your thoughts had been
02
IV DEDICATION.
more free, to take a leading share in that Reform of Philosophy,
which all who are alive to such errours, must see to be now in
dispensable. To you I may most justly inscribe a. work which
contains a criticism of the fallacies of the ultra-Lockian school.
I will mention one other reason which enters into the satisfac
tion with which I place your name at the head of my Philosophy.
By doing so, I may consider myself as dedicating it to the College
to which we both belong, to which we both owe so much of all
that we arc, and in which we have lived together so long and so
happily; and that, be it remembered, the College of Bacon and of
Newton. That College, I know, holds a strong place in your affec
tions, as in mine ; and among many reasons, not least on this
account ; — we believe that sound and enduring philosophy ever
finds there a congenial soil and a fostering shelter. If the doc
trines which the present work contains be really true and valu
able, my unhesitating trust is, that they will spread gradually
from these precincts to every part of the land.
That this office of being the fosterer and diffuser of truth may
ever belong to our common Nursing Mother, and that you, my
dear Sedgwick, may long witness and contribute to these bene
ficial influences, is the hearty wish of
Yours affectionately,
W. WHEWELL.
Trinity College, May ^, 1840.
P KEF AC K
TO THE
SECOND EDITION.
1\ the Preface to the first edition of this work, it was
stated that the work was intended as an application of
the plan of Bacon's Novum Oryanon to the present conr
dition of Physical Science. Such an undertaking, it was
there said, plainly belongs to the present generation.
Bacon only divined how sciences might be constructed ;
we can trace, in their history, how their construction
has taken place. However sagacious were his conjec
tures, it may be expected that they will be further illus
trated by facts which we know to have really occurred.
However large were his anticipations, the actual progress
of science since his time may aid in giving comprehen
siveness to our views. And with respect to the methods
by which science is to be promoted, — the structure and
operation of the Oryan by which truth is to be collected
from nature, — we know that, though Bacon's general
maxims still guide and animate philosophical enquirers
yet that his views, in their detail, have all turned out
inapplicable : the technical parts of his method failed in
his hands, and are forgotten among the cultivators of
science. It cannot be an unfit task, at the present day,
to endeavour to extract from the actual past progress
of science, the elements of a more effectual and sub-
VI PREFACE TO
stantial Method of Discovery. The advances which
have, during the last three centuries, been made in the
physical sciences ; — in Astronomy, in Physics, in Che
mistry, in Natural History, in Physiology; — these are
allowed by all to be real, to be great, to be striking :
may it not be, then, that these steps of progress have
in them something alike? — that in each advancing move
ment there is some common process, some common prin
ciple ? — that the organ by which discoveries have been
made has had something uniform in its structure and
working ? If this be so, and if we can, by attending to
the past history of science, discover something of this
common element and common process in all discoveries,
we shall have a Philosophy of Science, such as our times
may naturally hope for : — we shall have the New Organ
of Bacon, renovated according to our advanced intellec
tual position and office.
It was with the view to such a continuation and
extension of Bacon's design, that I undertook that sur
vey of the History of Science which I have given in
another work ; and that analysis of the advance of each
science which the present work contains. Of the doc
trines promulgated by Bacon, none has more completely
remained with us, as a stable and valuable truth, than
his declaration that true knowledge is to be obtained
from Facts by Induction : and in order to denote that I
start at once from the point to which Bacon thus led us,
I have, both in the History and in the Philosophy, termed
the sciences with which I have to do, the Inductive Sci
ences. By treating of the Physical Sciences only, while
I speak of the Inductive Sciences in the description of
THE SECOND EDITION. vii
ray design, I do not, (as I have already elsewhere said*)
intend to deny the character of Inductive Sciences to
many other branches of knowledge, as for instance, Eth
nology, Glossology, Political Economy, and Psychology.
But I think it will be allowed that by taking, as I have
done, the Physical Sciences alone, in which the truths
established are universally assented to, and regarded with
comparative calmness, we are better able to discuss the
formal conditions and general processes of scientific
discovery, than we could do if we entangled ourselves
among subjects where the interest is keener and the
truth more controverted. Perhaps a more exact descrip
tion of the present work would be, The Philosophy of
the Inductive Sciences, founded upon the History of the
principal Physical Sciences.
I am well aware how much additional interest and
attractiveness are given to speculations concerning the
progress of human knowledge, when we include in them,
as examples of such knowledge, views on subjects of
politics, morals, beauty in art and literature, and the like.
Prominent instances of the effect of this mode of treating
such subjects have recently appeared. But I still think
that the real value and import of Inductive Philosophy,
even in its application to such subjects, are best brought
into view by making the progress of political, and moral
and callesthetical^ truth a subject of consideration apart
from physical science.
It can hardly happen that a work which treats of
Methods of Scientific Discovery shall not seem to fail in
* Hist. I ml. Sci. Second Edition. Note to the Introduction,
t Sec Vol. ii. On the Language of Science, Aphorism, xvn.
Vlll PREFACE TO
the positive results which it offers. For an Art of Dis
covery is not possible. At each step of the progress of
science, are needed invention, sagacity, genius ; — elements
which no Art can give. We may hope in vain, as Bacon
hoped, for an organ which shall enable all men to construct
scientific truths, as a pair of compasses enables all men
to construct exact circles *. The practical results of the
Philosophy of Science must, we are persuaded, be rather
classification and analysis than precept and method. 1
think however that the methods of discovery which
I have to recommend, though gathered from a wider
survey of scientific history, as to subject and as to
time, than, (so far as I am aware,) has been elsewhere
attempted, are quite as definite and practical as any
others which have been proposed ; with the great addi
tional advantage of being the methods by which all great
discoveries in physical science really have been made.
This may be said, for instance, of the Method of Grada
tion, and the Method of Natural Classification, spoken
of Book XIIL Chap. vm. ; and in a narrower sense, of
the Method of Curves, the Method of Means, the Method
of Least Squares, and the Method of Residues, spoken
of in Chap. vn. of the same Book. Also the Remarks
on the Use of Hypotheses and on the Tests of Hypotheses
(Book xi. Chap, v.) point out features which mark the
usual course of discoverv.
*>
But undoubtedly one of the principal lessons which
results from the views here given is that different
sciences may be expected to advance by different modes
of procedure, according to their present condition ; and
*' AV. Or*,. Lib. i. Apli. (il
THE SECOND EDITION. IX
that, in many of these sciences, an Induction per
formed by any of the methods just referred to, is not
the step which we may expect to see next made.
Several of the sciences may not be in a condition which
fits them for such a Colligation of Facts, (to use the
phraseology to which the succeeding analysis has led
me. See B. xi. C. i). The Facts may, at the present
time, require to be more fully observed, or the Idea by
which they are to be colligated may require to be more
fully unfolded.
But in this point also, our speculations are far from
being barren of practical results. The Philosophy of
each Science, as given in the present work, affords us
means of discerning whether that which is needed for
the further progress of the Science has its place in the
Observations, or in the Ideas, or in the union of the two.
If Observations be wanted, the Methods of Observation
given in Book xm. Chap. u. may be referred to ; if
those who are to make the next discoveries need, for
that purpose, a developement of their Ideas, the modes
in which such a developement has usually taken place
are treated of in Chapters in. and iv. of that Book.
Perhaps one of the most prominent points of this
work is the attempt to show the place which discussions
concerning Ideas have had in the progress of science.
The metaphysical aspect of each of the physical sciences
is very far from being, as some have tried to teach, an
aspect which it passes through previously to the most
derided progress of the science. On the contrary, the
metaphysics] is a necessary part of the inductive move
ment. This, which is evidently so by the nature of the
X PREFACE TO
case, is proved by a copious collection of historical evi
dences in the first ten Books of the present work. Those
Books contain an account of the principal philosophical
controversies which have taken place in all the physical
sciences, from Mathematics to Physiology ; and these
controversies, which must be called metaphysical if any
thing be so called, have been conducted by the greatest
discoverers in each science, and have been an essential
part of the discoveries made. Physical discoverers have
differed from barren speculators, not by having no meta
physics in their heads, but by having good metaphysics
while their adversaries had bad ; and by binding their
metaphysics to their physics, instead of keeping the two
asunder. I trust that the ten Books of which I have
spoken are of some value, even as a series of analyses of
a number of remarkable controversies ; but I cannot con
ceive how any one, after reading these Books, can fail
to see that there is in progressive science a metaphysical
as well as a physical element ; — ideas, as well as facts,—
thoughts, as w^ell as things : — in short, that the Funda
mental Antithesis, for which I contend, is there most
abundantly and strikingly exemplified.
On the subject of this doctrine of a Fundamental
Analysis, which our knowledge always involves, I will
venture here to add a remark, which looks beyond the
domain of the physical sciences. This doctrine is suited
to throw light upon Moral and Political Philosophy, no
less than upon Physical. In Morality, in Legislation, in
National Polity, we have still to do with the opposition
and combination of two Elements ; — of Facts and Ideas ;
of Tlistorv, and an Ideal Standard of Action ; of actual
THE SECOND EDITION. xi
character and position, and of the aims which are placed
above the Actual. Each of these is in conflict with the
other ; each modifies and moulds the other. We can never
escape the control of the first ; we must .ever cease to
strive to extend the sway of the second. In these cases,
indeed, the Ideal Element assumes a new form. It in
cludes the Idea of Duty. The opposition, the action
and re-action, the harmony at which we must ever
aim, and can never reach, are between what is and what
ought to be ; — between the past or present Fact, and
the Supreme Idea. The Idea can never be independ
ent of the Fact, but the Fact must ever be drawn
towards the Idea. The History of Human Societies,
and of each Individual, is by the moral philosopher,
regarded in reference to this Antithesis ; and thus both
Public and Private Morality becomes an actual progress
towards an Ideal Form ; or ceases to be a moral reality.
I have made very slight alterations in the first
edition, except that the First Book is remodelled with
a view of bringing out more clearly the basis of the
work ; — this doctrine of the Fundamental Antithesis of
Philosophy. This doctrine, and its relation to the rest
of the work, have become more clear in the years
which have elapsed since the first edition.
A separate Essay, in which this doctrine was ex
plained, and a few other Essays previously published in
various forms, and containing discussions of special
points belonging to the scheme of philosophy here de
livered, have attracted some notice, both in this and in
other countries. I have therefore added them as an
Appendix to the present edition.
Xll PREFACi; TO
I have added a few Notes, in answer to arguments
brought against particular parts of this work. I have
written these in what I have elsewhere called an im
personal manner ; wishing to avoid controversy, so far
as justice to philosophical Truth will allow me to do so.
I have not given any detailed reply to the criticisms
of this work which occur in Mr. Mill's System of Logic.
The consideration of these criticisms would be interest
ing to me, and I think would still further establish the
doctrines which I have here delivered. But such a dis
cussion would involve me in a critique of Mr. Mill's
work ; which if 1 were to offer to the world, I should
think it more suitable to publish separately.
More than one of my critics has expressed an opinion
that when I published this work, I had not given due at
tention to the Cours de Philosophic Positive of M. Comte.
I had, and have, an opinion of the value of M. Comte's
speculations very different from that entertained by my
monitors. I had in the former edition discussed, and,
as I conceive, confuted, some of M. Comte's leading-
doctrines*. In order further to show that I had not
lightly passed over those portions of M. Comte's work
which had then appeared, I now publish f an additional
portion of a critique of the work which, though I had
written, I excluded from the former edition. This is
printed exactly as it existed in manuscript at the
period of that publication. To return to the subject and
to take it up in all its extent, would be an undertaking
out of the range of a new edition of my published
work.
* B. xi. r. vii. B. \ni. c. iv. I I>. \n. c. xvi.
THE SECOND EDITION. Xlll
Bacon delivered his philosophy in Aphorisms ; — a
series of Sentences which profess to exhibit rather the
results of thought than the process of thinking. A
mere Aphoristic Philosophy unsupported by reasoning,
is not suited to the present time. No writer upon
such subjects can expect to be either understood or
assented to, beyond the limits of a narrow school, who
is not prepared with good arguments as well as magis
terial decisions upon the controverted points of philo
sophy. But it may be satisfactory to some readers to
see the Philosophy, to which in the present work we are
led, presented in the Aphoristic form. I have therefore
placed a Series of Aphorisms at the end of the work.
In the former edition these, by being placed at the begin
ning of the work, might mislead the reader ; seeming
to some, perhaps, to be put forwards as the grounds, not
as the results, of our philosophy. I have also prefixed
an analysis of the work, in the form of a Table of Con
tents to each volume.
In that part of the second volume which treats of
the Language of Science, I have made a few alterations
and additions, tending to bring my recommendations
into harmony with the present use of the best scientific
works.
CONTENTS
THE FIRST VOLUME.
PREFACE
PART I.
OF IDEAS.
BOOK I.
OF IDEAS IN GENERAL.
CHAP. I. INTRODUCTION 1
CHAP. II. OF THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY . 1(5
Sect. 1. Thoughts and Things.
2. Necessary and Experiential Truths • 19
3. Deduction and Induction . . . . .21
4. Theories and Facts 23
5. Ideas and Sensations ...... 24
6. Reflexion and Sensation ..... 27
7- Subjective and Objective ... . 29
8. Matter and Form 33
9. Man the Interpreter of Nature . . . 37
10. The Fundamental Antithesis is inseparable . . 38
11. Successive Generalization . . . . 46.
CHAP. III. OF TECHNICAL TERMS . . . . . 51
Art. 1. Examples.
2. Use of Terms.
CHAP. IV. OF NECESSARY TRUTHS 54
Art. 1. The two Elements of Knowledge,
2. Shewn by necessary Truths.
3. Examples of necessary Truths in numbers.
4. The opposite cannot be distinctly conceived.
5. Other Examples.
6. Universal Truths.
CHAP. V. OF EXPERIENCE . . t>2
Art. 1. Experience cannot prove necessary Truths,
2. Except when aided by Ideas.
XVI CONTENTS OF
PAGE
CHAT. VI. OF THE GROUNDS OF NECESSARY TRUTHS . (50
Art, 1. These Grounds are Fundamental Ideas.
2. These are to be reviewed.
3. Definitions and Axioms.
4. Syllogism,
f). Produces no new Truths.
(i. Axioms needed.
7. Axioms depend on Ideas :
8. So do Definitions.
9. Idea not completely expressed.
CHAP. VII. THE FUNDAMENTAL IDEAS ARE NOT DERIVED FROM
EXPERIENCE . . 74
Art. 1. No connexion observed.
2. Faculties implied in observation.
3. We arc to examine our Faculties.
CHAP. VIII. OF THE PHILOSOPHY OF THE SCIENCES . . 7^
Sciences arranged according to Ideas.
BOOK II.
THE PHILOSOPHY OF THE PURE SCIENCES.
CHAP. I. OF THE PURE SCIENCES ..... 82
Art. I. Geometry, Arithmetic, Algebra,
2. Are not Inductive Sciences :
3. Are Mathematical Sciences.
4. Mixed Mathematics.
5. Space, Time, Number.
CHAP. II. OF THE IDEA OF SPACE . 84
Art. 1. Space is an Idea,
2. Not derived from Experience,
3. As Geometrical Truth shews.
4. Space is a Form of Experience.
5. The phrase not essential.
CHAP. III. OF SOME PECULIARITIES OF THE IDEA OF SPACE' . 88
Art. 1. Space is not an Abstract Notion.
2. Space is infinite.
3. Space is real.
4. Space is a Form of Intuition.
f>. Figure.
0. Three Dimensions.
THE FIRST VOLUME. XV11
PAGE
CHAP. IV. OF THE DEFINITIONS AND AXIOMS WHICH RELATE TO
SPACE .91
Art. I. Geometry.
2. Definitions.
3. Axioms.
4. Not Hypotheses.
5. Axioms necessary.
6. Straight lines.
7- Planes.
8. Elementary Geometry.
CHAP. V. OF SOME OBJECTIONS WHICH HAVE BEEN MADE TO THE
DOCTRINES STATED IN THE PREVIOUS CHAPTER . 101
Art. 1. How is Geometry hypothetical?
2. What was Stewart's view ?
3. " Legitimate filiations " of Definitions.
4. Is a Definition a complete explanation ?
5. Are some Axioms Definitions ?
0. Axiom concerning Circles.
7. Can Axioms become truisms ? -
8. Use of such.
CHAP. VI. OF THE PERCEPTION OF SPACE . . . .111
Art. 1. Which Senses apprehend Space?
2. Perception of solid figure.
3. Is an interpretation.
4. May be analysed.
"). Outline.
6. Reversed convexity.
7- Do we perceive Space by Touch ?
8. Brown's Opinion.
9. The Muscular Sense.
10. Bell's Opinion.
11. Perception includes Activity.
12. Perception of the Skiey Dome.
13. Reid's Idomenians.
14. Motion of the Eye.
15. Searching Motion.
16. Sensible Spot.
17- Expressions implying Motion.
CHAP. VII. OF THE IDEA OF TIME . 12">
Art. 1. Time an Idea not derived from Experience.
2. Time is a Form of Experience.
VOL. I. W. P.
XV111 CONTENTS OF
PAGE
Art. 3. Number.
4. Is Time derived from Motion ?
CHAP. VIII. OF SOME PECULIARITIES IN THE IDEA OF TIME . . 128
Art. 1. Time is not an Abstract Notion.
2. Time is infinite.
3. Time is a Form of Intuition.
4. Time is of one Dimension,
5. And no more.
6. Rhythm.
7- Alternation.
8. Arithmetic.
CHAP. IX. OF THE AXIOMS WHICH RELATE TO NUMBER . . 132
Art. I. Grounds of Arithmetic.
2. Intuition.
3. Arithmetical Axioms,
4. Are Conditions of Numerical Reasoning
5. In all Arithmetical Operations.
6. Higher Numbers.
CHAP. X. OF THE PERCEPTION OF TIME AND NUMBER . . 135
Art. 1. Memory.
2. Sense of Successiveness
3. Implies Activity.
4. Number also does so.
5. And apprehension of Rhythm.
Note to Chapter X . . . ... . . .139
CHAP. XI. OF MATHEMATICAL REASONING . . . 141
Art. 1. Discursive Reasoning.
2. Technical Terms of Reasoning.
3. Geometrical Analysis and Synthesis.
CHAP. XII. OF THE FOUNDATIONS OF THE HIGHER MATHEMATICS 145
Art. 1. The Idea of a Limit.
2. The use of General Symbols.
3. Connexion of Symbols and Analysis.
CHAP. XIII. THE DOCTRINE OF MOTION . . . . 150
Art. 1. Pure Mechanism.
2. Formal Astronomy.
CHAP. XIV. OF THE APPLICATION OF MATHEMATICS TO THE
INDUCTIVE SCIENCES . . . . .153
Art. 1. The Ideas of Space and Number are clear from the
first.
THE FIRST VOLUME XIX
PAOB
Art. 2. Their application in Astronomy.
3. Conic Sections, &c.
4. Arabian Numerals.
5. Newton's Lemmas.
6. Tides.
7. Mechanics.
8. Optics.
9. Conclusion.
BOOK III.
THE PHILOSOPHY OF THE MECHANICAL SCIENCES.
CHAP. I. OF THE MECHANICAL SCIENCES . . 164
CHAP. II. OF THE IDEA OF CAUSE . . . . .105
Art. 1. Not derived from Observation.
2. As appears by its use.
3. Cause cannot be observed.
4. Is Cause only constant succession ?
5. Other reasons.
CHAP. III. MODERN OPINIONS RESPECTING THE IDEA OF CAUSE . 701
Art. 1. Hume's Doctrine.
2. Stewart and Brown.
3. Kant.
4. Relation of Kant and Brown.
5. Axioms flow from the Idea.
6. The Idea implies activity in the Mind.
CHAP. IV. OF THE AXIOMS WHICH RELATE TO THE IDEA OF CAUSE 177
Art. 1. Causes are Abstract Conceptions.
2. First Axiom.
3. Second Axiom.
4. Limitation of the Second Axiom.
5. Third Axiom.
6. Extent of the Third Axiom.
CHAP. V. OF THE ORIGIN OF OUR CONCEPTIONS OF FORCE AND
MATTER . .... 185
Art. 1. Force.
2. Matter.
3. Solidity.
4. Inertia.
5. Application.
bZ
XX CONTENTS OF
PAOE
CHAP. VI. OF THE ESTABLISHMENT OF THE PRINCIPLES OF
STATICS .... .192
Art. 1. Object of the Chapter.
2. Statics and Dynamics.
3. Equilibrium.
4. Measure of Statical Forces.
5. The Center of Gravity.
(5. Oblique Forces.
7- Force acts at any point of its Direction.
8. The Parallelogram of Forces
O
9. Is a necessary Truth.
10. Center of Gravity descends.
11. Stevinus's Proof.
12. Principle of Virtual Velocities.
13. Fluids press equally.
14. Foundation of this Axiom.
CHAP. VII. OF THE ESTABLISHMENT OF THE PRINCIPLES OF
DYNAMICS ...... 215
Art. 1. History.
2. The First Law of Motion.
3. Gravity is a Uniform Force.
4. The Second Law of Motion.
5. The Third Law of Motion.
6. Action and Reaction in Moving Bodies.
7- D'Alembert's Principle.
8. Connexion of Statics and Dynamics.
9. Mechanical Principles grow more evident.
10. Controversy of the Measure of Force.
CHAP. VIII. OF THE PARADOX OF UNIVERSAL PROPOSITIONS
OBTAINED FROM EXPERIENCE .... 245
Art. 1. Experience cannot establish necessary Truths ;
2. But can interpret Axioms
3. Gives us the Matter of Truths.
4. Exemplifies Truths.
5. Cannot shake Axioms.
6. Is this applicable in other cases ?
CHAP. IX. OF THE ESTABLISHMENT OF THE LAW OF UNIVERSAL
GRAVITATION ...... 254
Art. 1. General course of the History.
2. Particulars as to the Law.
THE FIRST VOLUME. XXI
PACK
Art. 3. As to the Gravity of Matter.
4. Universality of the Law.
5. Is Gravity an essential quality ?
6. Newton's Rule of Philosophizing.
7- Hypotheses respecting Gravity.
8. Do Bodies act at a distance ?
CHAP. X. OF THE GENERAL DIFFUSION OF CLEAR MECHANICAL
IDEAS ... . . 2(>2
Art. 1 . Nature of the Process
2. Among the Ancients.
3. Kepler, &c.
4. Lord Monboddo, &c.
5. Schelling, &c.
(i. Common usage.
7. Effect of Phrases.
8. Contempt of Predecessors.
9. Less detail hereafter.
10. Mechanico-Chemical Sciences.
11. Secondary Mechanical Sciences.
Additional Note to Chapter IV. On the Axioms which relate to
the Idea of Cause 274
Additional Note to Chapter VI. Sect. 5. On the Center of Gravity 275
BOOK IV.
THE PHILOSOPHY OF THE SECONDARY MECHANICAL
SCIENCES.
CHAP. I. OF THE IDEA OF A MEDIUM AS COMMONLY EMPLOYED . 277
Art. 1. Of Primary and Secondary Qualities.
2. The Idea of Externality.
3. Sensation by a Medium.
4. Process of Perception of Secondary Qualities.
CUAP. II. ON PECULIARITIES IN THE PERCEPTIONS OF THE DIF
FERENT SENSES ... . 28(i
Art. I. Difference of Senses.
Sect. I. Prerogatives of Sight.
Art. 2. Position.
3. Distance.
XX11 CONTENTS OF
PAGE
Sect. II. Prerogatives of Hearing.
Art. 4. Musical Intervals.
5. Chords.
6. Rhythm.
Sect. III. The Paradoxes of Vision.
Art. 7- First Paradox.
8. Second Paradox.
9. The same for near Objects.
10. Objections answered.
Sect. IV. The Perception of Visible Figures.
Art. 11. Brown's Opinion.
CHAP. III. SUCCESSIVE ATTEMPTS AT THE SCIENTIFIC APPLICA
TION OF THE IDEA OF A MEDIUM . . . 307
Art. 1. Introduction.
2. Sound.
3. Light.
4. Heat.
CHAP. IV. OF THE MEASURE OF SECONDARY QUALITIES . 3.
Sect. I. Scales of Qualities in General.
Art. 1. Intensity.
2. Quantity and Quality.
Sect. II. The Musical Scale.
Art. 3. Musical Relations.
4. Musical Standard.
Sect. III. Scales of Colour.
Art. 5. The Prismatic Scale.
6. Newton's Scale.
7- Scales of Impure Colours.
8. Chromatometer.
Sect. IV. Scales of Light.
Art. 9. Photometer.
10. Cyanometer.
Sect. V. Scales of Heat.
Art. 11. Thermometers.
12. Their progress.
13. Fixed Points.
14. Concordance of Thermometers.
15. Natural Measure.
THE FIRST VOLUME. XX111
PAGE
Art. 16. Law of Cooling.
17. Theory of Exchanges.
18. Air Thermometer.
19. Theory of Heat.
20. Other Instruments.
Sect. VI. Scales of other Quantities.
Art. 21. Tastes and Smells.
22. Quality of Sounds.
23. Articulate Sounds.
24. Transition.
BOOK V.
OF THE PHILOSOPHY OF THE MECHANICO-CHEMICAL
SCIENCES.
CHAP. I. ATTEMPTS AT THE SCIENTIFIC APPLICATION OF THE IDEA
OF POLARITY ....... 345
Art. 1. Introduction of the Idea.
2. Magnetism.
3. Electricity.
4. Voltaic Electricity.
5. Light.
6. Crystallization.
7- Chemical Affinity.
8. General Remarks.
9. Like repels like.
CHAP. II. OF THE CONNEXION OF POLARITIES . . . 357
Art. 1. Different Polar Phenomena from one Cause.
2. Connexion of Magnetic and Electric Polarity.
3. Ampere's Theory.
4. Faraday's views.
5. Connexion of Electrical and Chemical Polarity.
6. Davy's and Faraday's views
7« Depend upon Ideas as well as Experiments.
8. Faraday's Anticipations.
9. Connexion of Chemical and Crystalline Polarities.
10. Connexion of Crystalline and Optical Polarities.
11. Connexion of Polarities in general.
12. Schelling's Speculations.
13. Hegel's vague notions.
14. Ideas must giiide Experiment.
XXIV CONTENTS OF
PA OB
BOOK VI.
THE PHILOSOPHY OF CHEMISTRY.
CHAP. I. ATTEMPTS TO CONCEIVE ELEMENTARY COMPOSITION . 3J6
Art. 1. Fundamental Ideas of Chemistry.
2. Elements.
3. Do Compounds resemble their Elements ?
4. The Three Principles.
5. A Modern Errour.
6. Are Compounds determined by the Figure of Ele
ments ?
7. Crystalline Form depends on Figure of Elements.
8. Are Compounds determined by Mechanical Attrac
tion of Elements ?
9. Newton's followers.
10. Imperfection of their Hypotheses.
CUAP. II. ESTABLISHMENT AND DEVELOPMENT OF THE IDEA OF /
CHEMICAL AFFINITY .... . . 38!
Art. 1. Early Chemists.
2. Chemical Affinity.
3. Affinity or Attraction ?
4. Affinity preferable.
5. Analysis is possible.
6. Affinity is Elective.
7- Controversy on this.
8. Affinity is Definite.
9. Are these Principles necessarily true ?
10. Composition determines Properties.
11. Comparison on this subject.
12. Composition determines Crystalline Form.
CHAP. III. OF THE IDEA OF SUBSTANCE .... 404
Art. 1. Indestructibility of Substance.
2. The Idea of Substance.
3. Locke's Denial of Substance.
4. Is all Substance heavy ?
CHAP. IV. APPLICATION OF THE IDEA OF SUBSTANCE IN CHK-
MISTKY . 412
Art. 1. A Body is Equal to its Elements.
2. Lavoisier.
3. Are there Imponderable Elements ?
THE FIRST VOLUME. XXV
I'AliK
Art. 4. Faraday's views.
5. Composition of Water.
6. Heat in Chemistry.
CHAP. V. THE ATOMIC THEORY . .421
Art. 1. The Theory on Chemical Grounds.
2. Hypothesis of Atoms.
3. Its Chemical Difficulties.
4. Grounds of the Atomic Doctrine.
5. Ancient Atomists.
(5. Francis Bacon.
7- Modern Atomists.
8. Arguments for and against.
I). Boscovich's Theory.
10. Molecular Hypothesis.
11. Poisson's Inference.
12. Wollaston's Argument.
13. Properties are Permanent.
BOOK VII.
THE PHILOSOPHY OF MORPHOLOGY, INCLUDING
CRYSTALLOGRAPHY.
CHAP. I. EXPLICATION OP THE IDEA OF SYMMETRY . 439
Art. 1. Symmetry what.
2. Kinds of Symmetry.
3. Examples in Nature.
4. Vegetables and Animals.
5. Symmetry a Fundamental Idea.
0. Result of Symmetry.
CHAP. II. APPLICATION OF THE IDEA OF SYMMETRY TO CRYSTALS 447
Art. 1. " Fundamental Forms."
2. Their use.
3. " Systems of Crystallization."
4. Cleavage.
5. Other Properties.
CHAP. III. SPECULATIONS FOUNDED UPON THE SYMMETRY OF
CRYSTALS ... . 452
Art. 1. Integrant Molecules
2. Difficulties of the Theory.
3. Merit of the Theory.
4. Wollaston's Hypothesis.
XXVI CONTENTS OF
PAGE
Art. 5. Maxim for such Hypotheses.
6. Dalton's Hypothesis.
7- Ampere's Hypothesis.
8. Difficulty of such Hypotheses.
9. Isomorphism.
BOOK VIII.
PHILOSOPHY OF THE CLASSIFICATORY SCIENCES.
CHAP. I. THE IDEA OF LIKENESS AS GOVERNING THE USE OP
COMMON NAMES ...... 466
Art. 1. Object of the Chapter.
2. Unity of the Individual.
3. Condition of Unity.
4. Kinds.
5. Not made by Definitions.
6. Condition of the Use of Terms.
7- Terms may have different Uses.
8. Gradation of Kinds.
9. Characters of Kinds.
10. Difficulty of Definitions.
11. " The Five Words."
CHAP. II. THE METHODS OP NATURAL HISTORY, AS REGULATED
BY THE IDEA OF LIKENESS 4"9
,. Sect. I. Natural History in General.
Art. 1. Idea of Likeness in Natural History.
2. Condition of its Use.
Sect. II. Terminology.
Art. 3. Meaning of the word.
The Plan of the System,
Its Meaning.
Latent Reference to Natural Affinity.
Natural Classes.
Artificial Classes.
Are Genera Natural?
Natural History and Mathematics.
Natural Groups given by Type, not by Definition.
Physiography.
Artificial and Natural Systems.
THE FIRST VOLUME. XXV11
I'AOE
Sect. IV. Methods of framing Natural Systems.
Art. 13. Method of Blind Trial.
14. Method of General Comparison.
Sect. V. Gradation of Groups.
Art. 15. Series of Subdivisions.
What is a Species ?
The words " Species" and " Genus."
Varieties. Races.
Sect. VI. Nomenclature.
Art. 19. Binary Nomenclature.
Sect. VII. Diagnosis.
Art. 20. Characteristick and Systematick.
CHAP. III. APPLICATION OF THE NATURAL HISTORY METHOD TO
MINERALOGY . * . . 512
Art. 1. Mohs's System.
2. His " Characteristick."
3. Mineral Species not yet well fixed.
4. Orders of Minerals.
5. Nomenclature of Minerals.
6. M. Necker's " Rcgne Mineral."
7- Inconvenience of taking a Chemical basis of Mineral
Systems.
8. Relation of Natural History and Chemistry.
9. What is a Mineralogical Individual ?
10. A well-formed Crystal is an Individual.
11. Not the Integrant Molecules,
12. Nor the Cleavage Forms.
13. Compound Crystals are not individuals.
14. Crystalline Forms are sufficiently complete for this.
15. Including aggregate Masses.
16. Do Artificial Crystals belong to Mineralogy ?
17. The Mineralogical Individual extends as far as
the same Crystalline Axes extend.
18. Artificial Crystals do belong to Mineralogy :
19. Cannot be excluded.
20. Species to be determined by the Crystalline Power.
21. Secondary Derivative Forms are Varieties:
22. Are not Species, as M. Necker holds.
XXV111 CONTENTS OF
PA OK
CHAP. IV. Or THE IDEA OF NATURAL AFFINITY 535
Art. 1. The Idea of Affinity
2. Is not to be made out by Arbitrary Eules.
3. Functions of Living tilings are many,
4. But all lead to the same arrangement.
5. This is Cuvier's principle :
(5. And Decandolle's.
7- Is this applicable to Inorganic Bodies ?
8. Yes ; by the agreement of Physical and Chemical
Arrangement.
BOOK IX.
THE PHILOSOPHY OF BIOLOGY.
CHAP. I. ANALOGY OF BIOLOGY \VITII OTHER SCIENCES . 543
Art. 1 . Biology involves the Idea of Life.
~».'
2. This Idea to be historically traced.
3. The Idea at first expressed by means of other Ideas,
4. Mystical, Mechanical, Chemical, and Vital Fluid
Hypotheses.
CHAP. II. SUCCESSIVE BIOLOGICAL HYPOTHESES . . . 548
Sect. I. The Mystical School.
Sect. II. The latrochemical School.
Sect. III. The latromathematical School.
Sect. IV. The Vitnl Fluid School.
Sect. V. The Psychical School.
CHAP. III. ATTEMPTS TO ANALYSE THE IDEA OF LIFE . . 571
A rt. } . Definitions of Life,
2. By Stahl, Humboldt, Kant.
3. Definition of Organization by Kant.
4. Life is a System of Functions.
5. Bichat. Sum of Functions.
<). Use of Definition.
7- Cuvier's view.
8. Classifications of Functions.
0. Vital, Natural, and Animal Functions.
10. Bichat. Organic and Animal Life.
1 1 . Use of this Classification.
THE FIRST VOLUME. XXIX
PACK
CHAP. IV. ATTEMPTS TO FORM IDEAS op SEPARATE VITAL
FORCES, AND FIRST, OF ASSIMILATION AND SECRE
TION 580
Sect. I. Course of Biological Research.
Art. 1. Observation and New Conceptions.
Sect. II. Attempts to form a distinct Conception of Assimila
tion and Secretion.
Art. 2. The Ancients.
3. Buffon. Interior Mould.
4. Defect of this view.
5. Cuvier. Life a Vortex.
6. Defect of this view.
7. Schclling. Matter and Form.
8. Life a constant Form of circulating Matter, &c.
Sect. III. Attempts to conceive the Forces of Assimilation and
Secretion.
Art. 9. Assimilation is a Vital Force.
10. The name "Assimilation."
11. Several processes involved in Assimilation.
12. Absorption. Endosmose.
13. Absorption involves a Vital Force.
14. Secretion. Glands.
15. Motions of Vital Fluids.
Sect. IV. Attempts to conceive the Process of Generation.
Art. 16. Reproduction figuratively used for Generation.
1 7- Nutrition different from
18. Generation.
19. Generations successively included.
20. Pre-existence of Germs.
21. Difficulty of this view.
22. Communication of Vital Forces.
23. Close similarity of Nutrition and Generation.
24. The Identity of the two Processes exemplified.
CHAP. V. ATTEMPTS TO FORM IDEAS OF SEPARATE VITAL FORCES,
continued. — VOLUNTARY MOTION . . . (iOO
Art. 1. Voluntary Motion one of the animal Functions.
2. Progressive knowledge of it.
3. Nervous Fluid not electric.
4. Irritability. Glisson.
5. Ilaller.
XXX CONTENTS OF
PAGB
Art. 6. Contractility.
7. Organic Sensibility and Contractility not separable.
8. Improperly described by Bichat.
9. Brown.
10. Contractility a peculiar Power.
11. Cuvier's view.
12. Elementary contractile Action.
13. Strength of Muscular Fibre.
14. Sensations become Perceptions
15. By means of Ideas ;
16. And lead to Muscular Actions.
17- Volition comes between Perception and Action.
18. Transition to Psychology.
19. A center is introduced.
20. The central consciousness may be obscure.
21. Reflex Muscular Action.
22. Instinct.
23. Difficulty of conceiving Instinct.
24. Instinct opposed to Insight.
CHAP. VI. OF THE IDEA OF FINAL CAUSES . . . 618
Art. 1. Organization. Parts are Ends and Means.
2. Not merely mutually dependent.
3. Not merely mutually Cause and Effect.
4. Notion of End not derived from Facts.
5. This notion has regulated Physiology.
6. Notion of Design comes from within.
7- Design not understood by Savages.
8. Design opposed to Morphology.
9. Impression of Design when fresh.
10. Acknowledgement of an End by adverse Physiolo
gists.
1 1 . This included in the Notion of Disease.
12. It belongs to Organized Creatures only.
13. The term Final Cause-
14. Law and Design.
15. Final Causes and Morphology.
16. Expressions of physiological Ends.
17- The Conditions of Existence.
18. The asserted presumption of Teleology.
19. Final Causes in other subjects.
20. Transition to Palaetiology.
THE FIRST VOLUME. XXXI
PACK
BOOK X.
THE PHILOSOPHY OF PAL.CTIOLOGY.
CHAP. I. OF PALETIOLOGICAL SCIENCES IN GENERAL . . C37
Art. 1. Description of Paltetiology.
2. Its Members.
3. Other Members.
4. Connexion of the whole subject.
5. We shall take Material Sciences only;
6. But these are connected with others.
CIIAI'. II. OF THE THREE MEMBERS OF A PAL^ETIOLOGICAL
SCIENCE ........ 642
Art. 1. Divisions of such Sciences.
2. The Study of Causes.
3. vEtiology.
4. Phenomenology requires Classification. Phenomenal
Geology.
5. Phenomenal Uranology.
6. Phenomenal Geography of Plants and Animals.
7- Phenomenal Glossology.
8. The Study of Phenomena leads to Theory.
9. No sound Theory without ^Etiology.
10. Causes in Palietiology.
11. Various kinds of Cause.
12. Hypothetical Order of Palsetiological Causes.
13. Mode of Cultivating ./Etiology : — In Geology :
14. In the Geography of Plants and Animals :
15. In Languages.
16. Construction of Theories.
17. No sound Paleetiological Theory yet extant.
CHAP. III. OF THE DOCTRINE OF CATASTROPHES AND THE DOC
TRINE OF UNIFORMITY . ... 665
Art. 1. Doctrine of Catastrophes.
2. Doctrine of Uniformity.
3. Is Uniformity probable a priori ?
4. Cycle of Uniformity indefinite.
5. Uniformitarian Arguments are Negative only.
6. Uniformity in the Organic World.
7- Origin of the present Organic World.
8. Nebular Origin of the Solar System.
9. Origin of Languages.
10. No Natural Origin discoverable.
XXX11 CONTENTS OF THE FIRST VOLUME.
PAGE
CHAP. IV. OP THE RELATION or TRADITION TO PAL^TIOLOGY 680
Art. 1. Importance of Tradition.
2. Connexion of Tradition and Science.
3. Natural and Providential History of the World.
4. The Sacred Narrative.
5. Difficulties in interpreting the Sacred Narrative,
(x Such Difficulties inevitable.
7. Science tells us nothing concerning Creation.
8. Scientific views, when familiar, do not disturb the
authority of Scripture.
9. When should Old Interpretations be given up ?
10. In what Spirit should the Change be accepted ?
11. In what Spirit should the Change be urged?
12. Duty of Mutual Forbearance.
13. Case of Galileo.
CHAP. V. OF THE CONCEPTION OF A FIRST CAUSE . . . 700
Art. I. The Origin of things is not naturally discoverable;
2. Yet has always been sought after.
3. There must be a First Cause.
4. This is an Axiom.
5. Involved in the Proof of a Deity.
(?. The Mind is not satisfied without it.
7- The Whole Course of Nature must have a Cause.
8. Necessary Existence of God.
9. Forms of the Proof.
10. Idea of a First Cause is Necessary.
11. Conception of a First Cause.
12. The First Cause in all Sciences is the same.
13. We are thus led to Moral Subjects.
Conclusion of Part I.
THE
PHILOSOPHY
OF THE
INDUCTIVE SCIENCES.
PART I.
OF IDEAS.
VOL. I. W. P.
l^uu,' adhuc inventa sunt in Scieutiis, ea hujusiaodi sunt
nt Notionibus Vnlgaribus fere subjaceant : ut vero ad inte-
riora ct remotiora Nature pcnetrctur, uecesse cst ut tarn
NoxiOiNEs quani AXIOMATA inagis certa et nninita \ia a
particularibus abstrahantur ; atque omnino molior ct certior
iiitellcctus adoperatio in usum veniat.
BACON, Nov. Org., Lib. i. Aphor. xviii.
BOOK I.
OF IDEAS IN GENERAL.
CHAPTER I.
INTRODUCTION.
THE PHILOSOPHY OF SCIENCE, if the phrase were to be
understood in the comprehensive sense which most na
turally offers itself to our thoughts, would imply nothing
less than a complete insight into the essence and con
ditions of all real knowledge, and an exposition of the
best methods for the discovery of new truths. We must
narrow and lower this conception, in order to mould it
into a form in which we may make it the immediate
object of our labours with a good hope of success ; yet
still it may be a rational and useful undertaking, to
endeavour to make some advance towards such a Philo
sophy, even according to the most ample conception
of it which we can form. The present work has been
written with a view of contributing, in some measure,
however small it may be, towards such an undertaking.
But in this, as in every attempt to advance beyond
the position which we at present occupy, our hope of
success must depend mainly upon our being able to
profit, to the fullest extent, by the progress already
made. We may best hope to understand the nature and
conditions of real knowledge, by studying the nature
and conditions of the most certain and stable portions of
knowledge which we already possess : and we are most
likely to learn the best methods of discovering truth, by
VOL. i. w. P. B
2 OF IDEAS IN GENERAL.
examining how truths, now universally recognized, have
really been discovered. Now there do exist among us
doctrines of solid and acknowledged certainty, and
truths of which the discovery has been received with
universal applause. These constitute what we com
monly term Sciences ; and of these bodies of exact and
enduring knowledge, we have within our reach so large
and varied a collection, that we may examine them, and
the history of their formation, with a good prospect of
deriving from the study such instruction as we seek.
We may best hope to make some progress towards the
Philosophy of Science, by employing ourselves upon THE
PHILOSOPHY OF THE SCIENCES.
The Sciences to which the name is most commonly
and unhesitatingly given, are those which are concerned
about the material world ; whether they deal with the
celestial bodies, as the sun and stars, or the earth and
its products, or the elements ; whether they consider the
differences which prevail among such objects, or their
origin, or their mutual operation. And in all these
Sciences it is familiarly understood and assumed, that
their doctrines are obtained by a common process of
collecting general truths from particular observed facts,
which process is termed Induction. It is further assumed
that both in these and in other provinces of knowledge,
so long as this process is duly and legitimately per
formed, the results will be real substantial truth. And
although this process, with the conditions under which
it is legitimate, and the general laws of the formation of
Sciences, will hereafter be subjects of discussion in this
work, I shall at present so far adopt the assumption of
which I speak, as to give to the Sciences from which
our lessons are to be collected the name of Inductive
Sciences. And thus it is that I am led to designate my
work as THE PHILOSOPHY OF THE INDUCTIVE SCIENCES.
INTRODUCTION. 3
The views respecting the nature and progress of
knowledge, towards which we shall be directed by such
a course of inquiry as I have pointed out, though derived
from those portions of human knowledge which are
more peculiarly and technically termed Sciences, will by
no means be confined, in their bearing, to the domain of
such Sciences as deal with the material world, nor even
to the whole range of Sciences now existing. On the
contrary, we shall be led to believe that the nature of
truth is in all subjects the same, and that its discovery
involves, in all cases, the like conditions. On one sub
ject of human speculation after another, man's know
ledge assumes that exact and substantial character which
leads us to term it Science ; and in all these cases, whe
ther inert matter or living bodies, whether permanent
relations or successive occurrences, be the subject of our
attention, we can point out certain universal characters
which belong to truth, certain general laws which have
regulated its progress among men. And we naturally
expect that, even when we extend our range of specu
lation wider still, when we contemplate the world within
us as well as the world without us, when we consider
the thoughts and actions of men as well as the motions
and operations of unintelligent bodies, we shall still find
some general analogies which belong to the essence of
truth, and run through the whole intellectual universe.
Hence we have reason to trust that a just Philosophy of
the Sciences may throw light upon the nature and extent
of our knowledge in every department of human specu
lation. By considering what is the real import of our
acquisitions, where they are certain and definite, we may
learn something respecting the difference between true
knowledge and its precarious or illusory semblances ; by
examining the steps by which such acquisitions have
been made, we may discover the conditions under which
B 12
4 OF IDEAS IN GENERAL.
truth is to be obtained ; by tracing the boundary-line
between our knowledge and our ignorance, we may
ascertain in some measure the extent of the powers of
man's understanding.
But it may be said, in such a design there is nothing
new; these are objects at which inquiring men have
often before aimed. To determine the difference be
tween real and imaginary knowledge, the conditions
under which we arrive at truth, the range of the powers
of the human mind, has been a favourite employment of
speculative men from the earliest to the most recent
times. To inquire into the original, certainty, and com
pass of man's knowledge, the limits of his capacity, the
strength and weakness of his reason, has been the pro
fessed purpose of many of the most conspicuous and
valued labours of the philosophers of all periods up to
our own day. It may appear, therefore, that there is
little necessity to add one more to these numerous
essays ; and little hope that any new attempt will make
any very important addition to the stores of thought
upon such questions, which have been accumulated by
the profoundest and acutest thinkers of all ages.
To this I reply, that without at all disparaging the
value or importance of the labours of those who have
previously written respecting the foundations and con
ditions of human knowledge, it may still be possible to
add something to what they have done. The writings of
all great philosophers, up to our own time, form a series
which is not yet terminated. The books and systems of
philosophy which have, each in its own time, won the
admiration of men, and exercised a powerful influence
upon their thoughts, have had each its own part and
functions in the intellectual history of the world ; and
other labours which shall succeed these may also have
their proper office and useful effect. We may not be
INTRODUCTION. 5
able to do much, and yet still it may be in our power to
effect something. Perhaps the very advances made by
former inquirers may have made it possible for us, at
present, to advance still further. In the discovery of
truth, in the developement of man's mental powers and
privileges, each generation has its assigned part ; and it
is for us to endeavour to perform our portion of this
perpetual task of our species. Although the terms
which describe our undertaking may be the same which
have often been employed by previous writers to express
their purpose, yet our position is different from theirs,
and thus the result mav be different too. We have, as
•/
they had, to run our appropriate course of speculation
with the exertion of our best powers ; but our course
lies in a more advanced part of the great line along
which Philosophy travels from age to age. However
familiar and old, therefore, be the design of such a work
as this, the execution may have, and if it be performed
in a manner suitable to the time, will have, something
that is new and not unimportant.
Indeed, it appears to be absolutely necessary, in
order to check the prevalence of grave and pernicious
errour, that the doctrines which are taught concerning
the foundations of human knowledge and the powers of
the human mind, should be from time to time revised
and corrected or extended. Erroneous and partial views
are promulgated and accepted ; one portion of the truth
is insisted upon to the undue exclusion of another ; or
principles true in themselves are exaggerated till they
produce on men's minds the effect of falsehood. When
evils of this kind have grown to a serious height, a
Reform is requisite. The faults of the existing systems
must be remedied by correcting what is wrong, and sup
plying what is wanting. In such cases, all the merits
and excellencies of the labours of the preceding times do
6 OF IDEAS IN GENERAL.
not supersede the necessity of putting forth new views
suited to the emergency which has arrived. The new
form which errour has assumed makes it proper to
endeavour to give a new and corresponding form to
truth. Thus the mere progress of time, and the natural
growth of opinion from one stage to another, leads to
the production of new systems and forms of philosophy.
It will be found, I think, that some of the doctrines now
most widely prevalent respecting the foundations and
nature of truth are of such a kind that a Reform is
needed. The present age seems, by many indications, to
be called upon to seek a sounder Philosophy of Know
ledge than is now current among us. To contribute
towards such a Philosophy is the object of the present
work. The work is, therefore, like all works which
take into account the most recent forms of speculative
doctrine, invested with a certain degree of novelty in its
aspect and import, by the mere time and circumstances
of its appearance.
But, moreover, we can point out a very important
peculiarity by which this work is, in its design, distin
guished from preceding essays on like subjects ; and this
difference appears to be of such a kind as may well
entitle us to expect some substantial addition to our
knowledge as the result of our labours. The peculiarity
of which I speak has already been announced ; — it is
this : that we purpose to collect our doctrines concerning
the nature of knowledge, and the best mode of acquiring
it, from a contemplation of the Structure and History of
those Sciences (the Material Sciences), which are univer
sally recognized as the clearest and surest examples of
knowledge and of discovery. It is by surveying and
studying the whole mass of such Sciences, and the
various steps of their progress, that we now hope to
approach to the true Philosophy of Science.
INTRODUCTION. 7
Now this, I venture to say, is a new method of pur
suing the philosophy of human knowledge. Those who
have hitherto endeavoured to explain the nature of
knowledge, and the process of discovery, have, it is true,
often illustrated their views by adducing special exam
ples of truths which they conceived to be established,
and by referring to the mode of their establishment.
But these examples have, for the most part, been taken
at random, not selected according to any principle or
system. Often they have involved doctrines so pre
carious or so vague that they confused rather than eluci
dated the subject ; and instead of a single difficulty,—
What is the nature of Knowledge? these attempts at
illustration introduced two, — What was the true analysis
of the Doctrines thus adduced? and, — Whether they
might safely be taken as types of real Knowledge ?
This has usually been the case when" there have
been adduced, as standard examples of the formation of
human knowledge, doctrines belonging to supposed sci
ences other than the material sciences; doctrines, for
example, of Political Economy, or Philology, or Morals,
or the Philosophy of the Fine Arts. I am very far from
thinking that, in regard to such subjects, there are no
important truths hitherto established : but it would seem
that those truths which have been obtained in these
provinces of knowledge, have not yet been fixed by
means of distinct and permanent phraseology, and sanc
tioned by universal reception, and formed into a con
nected system, and traced through the steps of their
gradual discovery and establishment, so as to make them
instructive examples of the nature and progress of truth
in general. Hereafter we trust to be able to show that
the progress of moral, and political, and philological,
and other knowledge, is governed by the same laws as
that of physical science. But since, at present, the
8 OF IDEAS IN GENERAL.
former class of subjects are full of controversy, doubt,
and obscurity, while the latter consist of undisputed
truths clearly understood and expressed, it may be con
sidered a wise procedure to make the latter class of
doctrines the basis of our speculations. And on the
having taken this course, is, in a great measure, my
hope founded, of obtaining valuable truths which have
escaped preceding inquirers.
But it may be said that many preceding writers on
the nature and progress of knowledge have taken their
examples abundantly from the Physical Sciences. It
would be easy to point out admirable works, which have
appeared during the present and former generations, in
which instances of discovery, borrowed from the Phy
sical Sciences, are introduced in a manner most happily
instructive. And to the works in which this has been
done, I gladly give my most cordial admiration. But at
the same time I may venture to remark that there still
remains a difference between my design and theirs : and
that I use the Physical Sciences as exemplifications of
the general progress of knowledge in a manner very
materially different from the course which is followed in
works such as are now referred to. For the conclusions
stated in the present work, respecting knowledge and
discovery, are drawn from a connected and systematic
survey of the whole range of Physical Science and its
History ; whereas, hitherto, philosophers have contented
themselves with adducing detached examples of scientific
doctrines, drawn from one or two departments of science.
So long as we select our examples in this arbitrary and
limited manner, we lose the best part of that philosophi
cal instruction, which the sciences are fitted to afford
when we consider them as all members of one series,
and as governed by rules which are the same for all.
Mathematical and chemical truths, physical and physio-
INTRODUCTION. 9
logical doctrines, the sciences of classification and of
causation, must alike be taken into our account, in order
that we may learn what are the general characters of
real knowledge. When our conclusions assume so com
prehensive a shape that they apply to a range of sub
jects so vast and varied as these, we may feel some con
fidence that they represent the genuine form of universal
and permanent truth. But if our exemplification is of a
narrower kind, it may easily cramp and disturb our phi
losophy. We may, for instance, render our views of
truth and its evidence so rigid and confined as to be
quite worthless, by founding them too much on the con
templation of mathematical truth. We may overlook
some of the most important steps in the general course
of discovery, by fixing our attention too exclusively
upon some one conspicuous group of discoveries, as, for
instance, those of Newton. We may misunderstand the
nature of physiological discoveries, by attempting to
force an analogy between them and discoveries of me
chanical laws, and by not attending to the intermediate
sciences which fill up the vast interval between these
extreme terms in the series of material sciences. In
these and in many other ways, a partial and arbitrary
reference to the material sciences in our inquiry into
human knowledge may mislead us ; or at least may fail
to give us those wider views, and that deeper insight,
which should result from a systematic study of the whole
range of sciences with this particular object.
The design of the following work, then, is to form a
Philosophy of Science, by analyzing the substance and
examining the progress of the existing body of the sci
ences. As a preliminary to this undertaking, a survey
of the history of the sciences was necessary. This,
accordingly, I have already performed ; and the result
of the labour thus undertaken has been laid before the
public as a History of the I?nluctire Sciences.
10 OF IDEAS IX GEXERAL.
In that work I have endeavoured to trace the steps
by which men acquired each main portion of that know
ledge on which they now look with so much confidence
and satisfaction. The events which that History relates,
the speculations and controversies which are there de
scribed, and discussions of the same kind, far more
extensive, which are there omitted, must all be taken
into our account at present, as the prominent and
standard examples of the circumstances which attend
the progress of knowledge. With so much of real his
torical fact before us, we may hope to avoid such views
of the processes of the human mind as are too partial
and limited, or too vague and loose, or too abstract and
unsubstantial, to represent fitly the real forms of dis
covery and of truth.
Of former attempts, made with the same viewr of
tracing the conditions of the progress of knowledge, that
of Bacon is perhaps the most conspicuous : and his
labours on this subject were opened by his book on the
Advancement of Learning, which contains, among other
matter, a survey of the then existing state of knowledge.
But this review was undertaken rather with the object
of ascertaining in what quarters future advances were to
be hoped for, than of learning by what means they were
to be made. His examination of the domain of human
knowledge was conducted rather with the view of dis
covering what remained undone, than of finding out how
so much had been done. Bacon's survey was made for
the purpose of tracing the boundaries, rather than of
detecting the principles of knowledge. "I will now
attempt," he says*, "to make a general and faithful
perambulation of learning, with an inquiry what parts
thereof lie fresh and waste, and not improved and con
verted by the industry of man ; to the end that such a
plot made and recorded to memory, may both minister
* Advancement of Learning, 1>. i. p. 74.
INTRODUCTION. 1 1
light to any public designation, and also serve to excite
voluntary endeavours." Nor will it be foreign to our
scheme also hereafter to examine with a like purpose
the frontier-line of man's intellectual estate. But the
object of our perambulation in the first place, is not so
much to determine the extent of the field, as the sources
of its fertility. We would learn by what plan and rules
of culture, conspiring with the native forces of the boun
teous soil, those rich harvests have been produced which
fill our garners. Bacon's maxims, on the other hand,
respecting the mode in which he conceived that know
ledge was thenceforth to be cultivated, have little refer
ence to the failures, still less to the successes, which are
recorded in his Review of the learning of his time. His
precepts are connected with his historical views in a
slight and unessential manner. His Philosophy of the
Sciences is not collected from the Sciences which are
noticed in his survey. Nor, in truth, could this, at the
time when he wrote, have easily been otherwise. At
that period, scarce any branch of physics existed as a
science, except Astronomy. The rules which Bacon gives
for the conduct of scientific researches are obtained, as
it were, by divination, from the contemplation of sub
jects with regard to which no sciences as yet were. His
instances of steps rightly or wrongly made in this path,
are in a great measure cases of his own devising. He
could not have exemplified his Aphorisms by references
to treatises then extant, on the laws of nature ; for the
constant burden of his exhortation is, that men up to
his time had almost universally followed an erroneous
course. And however we may admire the sagacity with
which he pointed the way along a better path, we have
this great advantage over him ; — that we can interrogate
the many travellers who since his time have journeyed
on this road. At the present day, when we have under
12 OF IDEAS IX GENERAL.
our notice so many sciences, of such wide extent, so well
established ; a Philosophy of the Sciences ought, it must
seem, to be founded, not upon conjecture, but upon an
examination of many instances; — should not consist of
a few vague and unconnected maxims, difficult and
doubtful in their application, but should form a system
of which every part has been repeatedly confirmed and
verified.
This accordingly it is the purpose of the present
work to attempt. But I may further observe, that as
my hope of making any progress in this undertaking is
founded upon the design of keeping constantly in view
the whole result of the past history and present con
dition of science, I have also been led to draw my les
sons from my examples in a manner more systematic
and regular, as appears to me, than has been done by
preceding writers. Bacon, as I have just said, was led
to his maxims for the promotion of knowledge by the
sagacity of his own mind, with little or no aid from
previous examples. Succeeding philosophers may often
have gathered useful instruction from the instances of
scientific truths and discoveries which thev adduced, but
«/
their conclusions were drawn from their instances casu
ally and arbitrarily. They took for their moral any
which the story might suggest. But such a proceeding
as this cannot suffice for us, whose aim is to obtain a
consistent body of philosophy from a contemplation of
the whole of Science and its History. For our purpose
it is necessary to resolve scientific truths into their con
ditions and ingredients, in order that we may see in
what manner each of these has been and is to be pro
vided, in the cases which we may have to consider. This
accordingly is necessarily the first part of our task : — to
analyze Scientific Truth into its Elements. This attempt
will occupy the earlier portion of the present work ; and
INTRODUCTION. ] 3
will necessarily be somewhat long, and perhaps, in many
parts, abstruse and uninviting. The risk of such an
inconvenience is inevitable ; for the inquiry brings before
us many of the most dark and entangled questions in
which men have at any time busied themselves. And
even if these can now be made clearer and plainer than
of yore, still they can be made so only by means of men
tal discipline and mental effort. Moreover this analysis
of scientific truth into its elements contains much, both
in its principles and in its results, different from the
doctrines most generally prevalent among us in recent
times : but on that very account this analysis is an
essential part of the doctrines which I have now to lay
before the reader: and I must therefore crave his
indulgence towards any portion of it which may appear
to him obscure or repulsive.
There is another circumstance which may tend to
make the present work less pleasing than others on the
same subject, in the nature of the examples of human
knowledge to wrhich I confine myself; all my instances
being, as I have said, taken from the material sciences.
For the truths belonging to these sciences are, for the
most part, neither so familiar nor so interesting to the
bulk of readers as those doctrines which belong to some
other subjects. Every general proposition concerning
politics or morals at once stirs up an interest in men's
bosoms, which makes them listen with curiosity to the
attempts to trace it to its origin and foundation. Every
rule of art or language brings before the mind of culti
vated men subjects of familiar and agreeable thought,
and is dwelt upon with pleasure for its own sake, as well
as on account of the philosophical lessons which it may
convey. But the curiosity which regards the truths of
physics or chemistry, or even of physiology and astro
nomy, is of a more limited and less animated kind.
•< '
14 OF IDEAS IN GENERAL.
Hence, in the mode of inquiry which I have prescribed
to myself, the examples which I have to adduce will not
amuse and relieve the reader's mind as much as they
might do, if I could allow myself to collect them from
the whole field of human knowledge. They will have in
them nothing to engage his fancy, or to warm his heart.
I am compelled to detain the listener in the chilly air
of the external world, in order that we may have the
advantage of full daylight.
But although I cannot avoid this inconvenience, so
far as it is one, I hope it will be recollected how great
are the advantages which we obtain by this restriction.
We are thus enabled to draw all our conclusions from
doctrines which are universally allowed to be eminently
certain, clear, and definite. The portions of knowledge
to which 1 refer are well known, and well established
among men, Their names are familiar, their assertions
uncontested. Astronomy and Geology, Mechanics and
Chemistry, Optics and Acoustics, Botany and Physiology,
are each recognized as large and substantial collections
of undoubted truths. Men are wont to dwell with pride
and triumph on the acquisitions of knowledge which
have been made in each of these provinces ; and to speak
with confidence of the certainty of their results. And all
can easily learn in what repositories these treasures of
human knowledge are to be found. When, therefore,
we begin our inquiry from such examples, we proceed
upon a solid foundation. With such a clear ground of
confidence, we shall not be met with general assertions
of the vagueness and uncertainty of human knowledge ;
with the question, What truth is, and How we are to
recognize it ; with complaints concerning the hopeless
ness and unprofitableness of such researches. We have,
at least, a definite problem before us. We have to
examine the structure and scheme, not of a shapeless
INTRODUCTION. 15
mass of incoherent materials, of which we doubt whether
it be a ruin or a natural wilderness, but of a fair and
lofty palace, still erect and tenanted, where hundreds of
different apartments belong to a common plan, where
every generation adds something to the extent and mag
nificence of the pile. The certainty and the constant
progress of science are things so unquestioned, that we
are at least engaged in an intelligible inquiry, when we
are examining the grounds and nature of that certainty,
the causes and laws of that progress.
To this enquiry, then, we now proceed. And in
entering upon this task, however our plan or our prin
ciples may differ from those of the eminent philosophers
who have endeavoured, in our own or in former times,
to illustrate or enforce the philosophy of science, we
most willingly acknowledge them as in many things our
leaders and teachers. Each reform must involve its own
peculiar principles, and the result of our attempts, so
far as they lead to a result, must be, in some respects,
different from those of former works. But we may still
share with the great writers who have treated this
subject before us, their spirit of hope and trust, their
reverence for the dignity of the subject, their belief in
the vast powers and boundless destiny of man. And we
may once more venture to use the words of hopeful
exhortation, with which the greatest of those who have
trodden this path encouraged himself and his followers
when he set out upon his way.
" Concerning ourselves we speak not ; but as touch
ing the matter which we have in hand, this we ask;—
that men deem it not to be the setting up an Opinion,
but the performing of a Work : and that they receive
this as a certainty ; that we are not laying the founda
tions of any sect or doctrine, but of the profit and
dignity of mankind. Furthermore, that being well dis-
10 OF IDEAS IN GENERAL.
posed to what shall advantage themselves, and putting
off factions and prejudices, they take common counsel
with us, to the end that being by these our aids and
appliances freed and defended from wanderings and
impediments, they may lend their hands also to the
labours which remain to be performed : and yet further,
that they be of good hope ; neither imagine to them
selves this our Reform as something of infinite dimen
sion, and beyond the grasp of mortal man, when in truth
it is the end and true limit of infinite errour ; and is by
no means unmindful of the condition of mortality and
humanity, not confiding that such a thing can be carried
to its perfect close in the space of one single age, but
assigning it as a task to a succession of generations."
CHAPTER II.
OF THE FUNDAMENTAL ANTITHESIS OF
PHILOSOPHY.
SECT. 1. — Thoughts and Things.
IN order that we may do something towards determining
the nature and conditions of human knowledge, (which
I have already stated as the purpose of this work,) I
shall have to refer to an antithesis or opposition, which
is familiar and generally recognized, and in which the
distinction of the things opposed to each other is com
monly considered very clear and plain. I shall have to
attempt to make this opposition sharper and stronger
than it is usually conceived, and yet to shew that the
distinction is far from being so clear and definite as it is
usually assumed to be : I shall have to point the con
trast, yet shew that the things which are contrasted
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 17
cannot be separated : — I must explain that the anti
thesis is constant and essential, but yet that there is no
fixed and permanent line dividing its members. I may
thus appear, in different parts of my discussion, to be
proceeding in opposite directions, but I hope that the
reader who gives me a patient attention will see that
both steps lead to the point of view to which I wish to
lead him.
The antithesis or opposition of which I speak is
denoted, with various modifications, by various pairs of
terms : I shall endeavour to show the connexion of these
different modes of expression, and I will begin with that
form which is the simplest and most idiomatic.
The simplest and most idiomatic expression of the
antithesis to which I refer is that in which we oppose to
each other THINGS and THOUGHTS. The opposition is
familiar and plain. Our Thoughts are something which
belongs to ourselves ; something which takes place
within us; they are what me think ; they are actions of
our minds. Things, on the contrary, are something
different from ourselves and independent of us ; some
thing which is without us ; they are ; we see them,
touch them, and thus know that they exist ; but we do
not make them by seeing or touching them, as we make
our Thoughts by thinking them ; we are passive, and
Things act upon our organs of perception.
Now what I wish especially to remark is this : that
in all human KNOWLEDGE both Thoughts and Things are
concerned. In every part of my knowledge there must
i)e some thing about which I know, and an internal act
of me who know. Thus, to take simple yet definite parts
of our knowledge, if I know that a solar year consists of
365 days, or a lunar month of 30 days, I know' some
thing about the sun or the moon ; namely, that those
objects perform certain revolutions and go through cer-
VOL. i. \v. p. C
18 OF IDEAS IN GENERAL.
tain changes, in those numbers of days; but I count
such numbers and conceive such revolutions and changes
by acts of my own thoughts. And both these elements
of my knowledge are indispensable. If there were not
such external Things as the sun and the moon I could
not have any knowledge of the progress of time as
marked by them. And however regular were the mo
tions of the sun and moon, if I could not count their
appearances and combine their changes into a cycle, or
if I could not understand this when done by other men,
I could not know anything about a year or a month. In
the former case I might be conceived as a human being,
possessing the human powers of thinking and reckoning,
but kept in a dark world with nothing to mark the pro
gress of existence, The latter is the case of brute ani
mals, which see the sun and moon, but do not know how
many days make a month or a year, because they have
not human powers of thinking and reckoning.
The two elements which are essential to our know
ledge in the above cases, are necessary to human know
ledge in all cases. In all cases, Knowledge implies a
combination of Thoughts and Things. Without this
combination, it would not be Knowledge. Without
Thoughts, there could be no connexion ; without Things,
there could be no reality. Thoughts and Things are so
intimately combined in our Knowledge, that we do not
look upon them as distinct. One single act of the mind
involves them both ; and their contrast disappears in
their union.
But though Knowledge requires the union of these
two elements, Philosophy requires the separation of
them, in order that the nature and structure of Know
ledge may be seen. Therefore I begin by considering
this separation. And I now proceed to speak of another
way of looking at the antithesis of which I have spoken;
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 19
and which I may, for the reasons which I have just
mentioned, call the FUNDAMENTAL ANTITHESIS OF PHI
LOSOPHY.
SECT. 2. — Necessary and Experiential Truths.
MOST persons are familiar with the distinction of ne
cessary and contingent truths. The former kind are
Truths which cannot but be true; as that 19 and 11
make 30 ; — that parallelograms upon the same base and
between the same parallels are equal: — that all the
angles in the same segment of a circle are equal. The
latter are Truths which it happens (contingit] are true ;
but which, for any thing which we can see, might have
been otherwise ; as that a lunar month contains 30 days,
or that the stars revolve in circles round the pole. The
latter kind of Truths are learnt by experience, and hence
we may call them Truths of Experience, or, for the sake
of convenience, Experiential Truths, in contrast with
Necessary Truths.
Geometrical propositions are the most manifest ex
amples of Necessary Truths. All persons who have read
and understood the elements of geometry, know that the
propositions above stated (that parallelograms upon the
same base and between the same parallels are equal ;
that all the angles in the same segment of a circle are
equal,) are necessarily true ; not only they are true, but
they must le true. The meaning of the terms being
understood, and the proof being gone through, the truth
of the propositions must be assented to. We learn these
propositions to be true by demonstrations deduced from
definitions and axioms ; and when we have thus learnt
them, we see that they could not be otherwise. In the
same manner, the truths which concern numbers are
necessary truths: 19 and 11 not only do make 30, but
must make that number, and cannot make anything else.
20 OF IDEAS IN GENERAL.
In the same manner, it is a necessary truth that half the
sum of two numbers added to half their difference is
equal to the greater number.
It is easy to find examples of Experiential Truths ;—
propositions which we know to be true, but know by
experience only. We know, in this way, that salt will
dissolve in water ; that plants cannot live without light ;
— in short, we know in this way all that we do know
in chemistry, physiology, and the material sciences in
general. I take the Sciences as my examples of human
knowledge, rather than the common truths of daily life,
or moral or political truths ; because, though the latter
are more generally interesting, the former are much
more definite and certain, and therefore better starting-
points for our speculations, as I have already said. And
we may take elementary astronomical truths as the most
familiar examples of Experiential Truths in the domain
of science.
With these examples, the distinction of Necessary
and Experiential Truths is, I hope, clear. The former
kind, we see to be true by thinking about them, and see
that they could not be otherwise. The latter kind, men
could never have discovered to be true without looking
at them ; and having so discovered them, still no one will
pretend to say they might not have been otherwise. For
aught we can see, the astronomical truths which express
the motions and periods of the sun, moon and stars,
might have been otherwise. If we had been placed in
another part of the solar system, our experiential truths
respecting days, years, and the motions of the heavenly
bodies, would have been other than they are, as we
know from astronomy itself.
It is evident that this distinction of Necessary and
Experiential Truths involves the same antithesis which
we have already considered ; — the antithesis of Thoughts
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 21
and Things. Necessary Truths are derived from our own
Thoughts : Experiential Truths are derived from our
observation of Things about us. The opposition of
Necessary and Experiential Truths is another aspect of
the Fundamental Antithesis of Philosophy.
SECT. 3. — Deduction and Induction.
I HAVE already stated that geometrical truths are
established by demonstrations deduced from definitions
and axioms. The term Deduction is specially applied
to such a course of demonstration of truths from defini
tions and axioms. In the case of the parallelograms
upon the same base and between the same parallels, we
prove certain triangles to be equal, by supposing them
placed so that their two bases have the same extremi
ties; and hence, referring to an Axiom respecting straight
lines, we infer that the bases coincide. We combine
these equal triangles with other equal spaces, and in this
way make up both the one and the other of the paral
lelograms, in such a manner as to shew that they are
equal. In this manner, going on step by step, deducing
the equality of the triangles from the axiom, and the
equality of the parallelograms from that of the triangles,
we travel to the conclusion. And this process of suc
cessive deduction is the scheme of all geometrical proof.
We begin with Definitions of the notions which we reason
about, and with Axioms, or self-evident truths, respecting
these notions; and we get, by reasoning from these, other
truths which are demonstratively evident ; and from
these truths again, others of the same kind, and so on.
We begin with our own Thoughts, which supply us with
Axioms to start from; and we reason from these, till we
rome to propositions which are applicable to the Things
about us; as for instance, the propositions respecting
circles and spheres are applicable to the motions of the
22 OF IDEAS IN GENERAL.
heavenly bodies. This is Deduction, or Deductive Rea
soning,
Experiential truths are acquired in a very different
way. In order to obtain such truths, we begin with
Things. In order to learn how many days there are in
a year, or in a lunar month, we must begin by observing
the sun and the moon. We must observe their changes
day by day, and try to make the cycle of change fit into
some notion of number which we supply from our own
Thoughts. We shall find that a cycle of 30 days nearly
will fit the changes of phase of the moon; — that a cycle
of 365 days nearly will fit the changes of daily motion
of the sun. Or, to go on to experiential truths of
which the discovery comes within the limits of the his
tory of science — we shall find (as Hipparchus found)
that the unequal motion of the sun among the stars,
such as observation shews it to be, may be fitly repre
sented by the notion of an eccentric; — a circle in which
the sun has an equable annual motion, the spectator not
being in the center of the circle. Again, in the same
manner, at a later period, Kepler started from more
exact observations of the sun, and compared them with
a supposed motion in a certain ellipse; and was able to
shew that, not a circle about an eccentric point, but an
ellipse, supplied the mode of conception which truly
agreed with the motion of the sun about the earth ; or
rather, as Copernicus had already shewn, of the earth
about the sun. In such cases, in which truths are ob
tained by beginning from observation of external things
and by finding some notion with which the Things, as
observed, agree, the truths are said to be obtained by
Induction. The process is an Inductive Process.
The contrast of the Deductive and Inductive process
is obvious. In the former, we proceed at each step
from general truths to particular applications of them ;
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 23
in the latter, from particular observations to a general
truth which includes them. In the former case we
may be said to reason downwards, in the latter case,
upwards; for general notions are conceived as stand
ing above particulars. Necessary truths are proved,
like arithmetical sums, by adding together the portions
of which they consist. An inductive truth is proved,
like the guess which answers a riddle, by its agreeing
with the facts described. Dcmonstation is irresistible
in its effect on the belief, but does not produce surprize,
because all the steps to the conclusion are exhibited,
before we arrive at the conclusion. Inductive infer
ence is not demonstrative, but it is often more striking
than demonstrative reasoning, because the intermediate
links between the particulars and the inference are not
shown. Deductive truths are the results of relations
among our own Thoughts. Inductive Truths are re
lations which we discern among existing Things; and
thus, this opposition of Deduction and Induction is again
an aspect of the Fundamental Antithesis already spoken
of.
SECT. 4. — Theories and Facts.
GENERAL experiential Truths, such as we have just
spoken of, are called Theories, and the particular
observations from which they are collected, and which
they include and explain, are called Facts. Thus Hip-
parchus's doctrine, that the sun moves in an eccentric
about the earth, is his Theory of the Sun, or the Eccen
tric Theory. The doctrine of Kepler, that the Earth
moves in an Ellipse about the Sun, is Kepler s Theory
of the Earth, the Elliptical Theory. Newton's doctrine
that this elliptical motion of the Earth about the Sun
is produced and governed by the Sun's attraction upon
the Earth, is the Newtonian theory, the Theory of
Attraction. Each of these Theories was accepted, be-
24 OF IDEAS IN GENERAL.
cause it included, connected and explained the Facts;
the Facts being, in the two former cases, the motions
of the Sun as observed; and in the other case, the ellip
tical motion of the Earth as known by Kepler's Theory.
This antithesis of Theory and Fact is included in what
has just been said of Inductive Propositions. A Theory
is an Inductive Proposition, and the Facts are the par
ticular observations from which, as I have said, such
Propositions are inferred by Induction. The Antithesis
of Theory and Fact implies the fundamental Antithesis
of Thoughts and Things;, for a Theory (that is, a true
Theory) may be described as a Thought which is con
templated distinct from Things and seen to agree with
them; while a Fact is a combination of our Thoughts
with Things in so complete agreement that we do not
regard them as separate.
Thus the antithesis of Theory and Fact involves the
antithesis of Thoughts and Things, but is not identical
with it. Facts involve Thoughts, for we know Facts only
by thinking about them. The Fact that the year consists
of 365 days; the Fact that the month consists of 30 days,
cannot be known to us, except we have the Thoughts
of Time, Number and Recurrence. But these Thoughts
are so familiar, that we have the Fact in our mind
as a simple Thing without attending to the Thought
which it involves. When we mould our Thoughts into a
Theory, we consider the Thought as distinct from the
Facts; but yet, though distinct, not independent of them;
for it is a true Theory, only by including and agreeing
with the Facts.
SECT. 5. — Ideas and Sensations.
WE have just seen that the antithesis of Theory and
Fact, although it involves the antithesis of Thoughts and
Things, is not identical with it. There are other modes
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 25
of expression also, which involve the same Fundamental
Antithesis, more or less modified. Of these, the pair of
words which in their relations appear to separate the
members of the antithesis most distinctly are Ideas and
Sensations. We see and hear and touch external things,
and thus perceive them by our senses; but in perceiving
them, we connect the impressions of sense according to
relations of space, time, number, likeness, cause, &c.
Now some at least of these kinds of connexion, as space,
time, number, may be contemplated distinct from the
things to which they are applied; and so contemplated,
I term them Ideas. And the other element, the impres
sions upon our senses which they connect, are called
Sensations.
I term space, time, cause, &c., Ideas, because they
are general relations among our sensations, apprehend
ed by an act of the mind, not by the senses simply.
These relations involve something beyond what the
senses alone could furnish. By the sense of sight we
see various shades and colours and shapes before us, but
the outlines by which they are separated into distinct
objects of definite forms, are the work of the mind itself.
And again, when we conceive visible things, not only as
surfaces of a certain form, but as solid bodies, placed at
various distances in space, we again exert an act of the
mind upon them. When we see a body move, we see
it move in a path or orbit, but this orbit is not itself
seen; it is constructed by the mind. In like manner
when we see the motions of a needle towards a mag
net, we do not see the attraction or force which pro
duces the effects; but we infer the force, by having in
our minds the Idea of Cause. Such acts of thought,
such Ideas, enter into our perceptions of external things.
But though our perceptions of external things in
volve some act of the mind, they must involve some-
26 OF IDEAS IN GENERAL.
thing else besides an act of the mind. If we must exer
cise an act of thought in order to see force exerted, or
orbits described by bodies in motion, or even in order
to see bodies existing in space, and to distinguish one
kind of object from another, still the act of thought
alone does not make the bodies. There must be some
thing besides, on mldcli the thought is exerted. A
colour, a form, a sound, are not produced by the mind,
however they may be moulded, combined, and inter
preted by our mental acts. A philosophical poet has
spoken of
All the world
Of eye and ear, both what they half create,
And what perceive.
But it is clear, that though they half create, they do not
wholly create: there must be an external world of colour
and sound to give impressions to the eye and ear, as
well as internal powers by which we perceive what is
offered to our organs. The mind is in some way passive
as well as active: there are objects without as well as
faculties within; — Sensations, as well as acts of Thought.
Indeed this is so far generally acknowledged, that
according to common apprehension, the mind is passive
rather than active in acquiring the knowledge which
it receives concerning the material world. Its sensa
tions are generally considered more distinct than its
operations. The world without is held to be more clearly
real than the faculties within. That there is some
thing different from ourselves, something external to us,
something independent of us, something which no act
of our minds can make or can destroy, is held by all
men to be at least as evident, as that our minds can
exert any effectual process in modifying and appreciating
the impressions made upon them. Most persons are
more likely to doubt whether the mind be always actively
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 27
applying Ideas to the objects which it perceives, than
whether it perceive them passively by means of Sen
sations.
But yet a little consideration will show us that an
activity of the mind, and an activity according to certain
Ideas, is requisite in all our knowledge of external
objects. We see objects, of various solid forms, and at
various distances from us. But we do not thus perceive
them by sensation alone. Our visual impressions can
not, of themselves, convey to us a knowledge of solid
form, or of distance from us. Such knowledge is inferred
from what we see : — inferred by conceiving the objects
as existing in space, and by applying to them the Idea of
Space. Again : — day after day passes, till they make up a
year : but we do not know that the days are 365, except
we count them; and thus apply to them our Idea of Num
ber. Again : — we see a needle drawn to a magnet : but,
in truth, the drawing is what we cannot see. We see the
needle move, and infer the attraction, by applying to the
fact our Idea of Force, as the cause of motion. Again: —
we see two trees of different kinds ; but we cannot know
that they are so, except by applying to them our Idea
of the resemblance and difference which makes kinds.
And thus Ideas, as well as Sensations, necessarily enter
into all our knowledge of objects : and these two words
express, perhaps more exactly than any of the pairs
before mentioned, that Fundamental Antithesis, in the
union of which, as I have said, all knowledge consists.
SECT 6. — Reflexion and Sensation.
IT will hereafter be my business to show what the
Ideas are, which thus enter into our knowledge ; and
how each Idea has been, as a matter of historical fact,
introduced into the Science to which it especially be
longs. But before I proceed to do this, I will notice
28 OF IDEAS IN GENERAL.
some other terms, besides the phrases already noticed,
which have a reference, more or less direct, to the Funda
mental Antithesis of Ideas and Sensations. I will mention
some of these, in order that if they should come under
the reader's notice, he may not be perplexed as to their
bearing upon the view here presented to him.
The celebrated doctrine of Locke, that all our
"Ideas," (that is, in his use of the word, all our objects
of thinking,) come from Sensation or Reflexion, will
naturally occur to the reader as connected with the
antithesis of which I have been speaking. But there is
a great difference between Locke's account of Sensation
and Reflexion, and our view of Sensation and Ideas. He
is speaking of the origin of our knowledge ; — we, of its
nature and composition. He is content to say that all
the knowledge which we do not receive directly by
Sensation, we obtain by Reflex Acts of the mind, which
make up his Reflexion. But we hold that there is no
Sensation without an act of the mind, and that the
mind's activity is not only reflexly exerted upon itself,
but directly upon objects, so as to perceive in them con
nexions and relations which are not Sensations. He is
content to put together, under the name of Reflexion,
everything in our knowledge which is not Sensation : we
are to attempt to analyze all that is not Sensation ; not
only to say it consists of Ideas, but to point out what
those Ideas are, and to show the mode in which each of
them enters into our knowledge. His purpose was, to
prove that there are no Ideas, except the reflex acts of
the mind : our endeavour will be to show that the acts of
the mind, both direct and reflex, are governed by certain
Laws, which may be conveniently termed Ideas. His
procedure was, to deny that any knowledge could be
derived from the mind alone : our course will be, to
show that in every part of our most certain and exact
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 29
knowledge, those who have added to our knowledge in
every age have referred to principles which the mind
itself supplies. I do not say that my view is contrary to
his : but it is altogether different from his. If I grant
that all our knowledge comes from Sensation and Re
flexion, still my task then is only begun ; for I want
further to determine, in each science, what portion
comes, not from mere Sensation, but from those Ideas
by the aid of which either Sensation or Reflexion can
lead to Science.
Locke's use of the word "idea" is, as the reader will
perceive, different from ours. He uses the word, as he
says, which " serves best to stand for whatsoever is the
object of the understanding when a man thinks." " I
have used it," he adds, " to express whatever is meant by
phantasm, notion, species, or whatever it is to which the
mind can be employed about in thinking." It might be
shown that this separation of the mind itself from the
ideal objects about which it is employed in thinking, may
lead to very erroneous results. But it may suffice to ob
serve that we use the word Ideas, in the manner already
explained, to express that element, supplied by the mind
itself, which must be combined with Sensation in order
to produce knowledge. For us, Ideas are not Objects of
Thought, but rather Laws of Thought. Ideas are not
synonymous with Notions; they are Principles which
give to our Notions whatever they contain of truth. But
our use of the term Idea will be more fully explained
hereafter.
SECT. 7 — Subjective and Objective.
THE Fundamental Antithesis of Philosophy of which I
have to speak has been brought into great prominence
in the writings of modern German philosophers, and has
conspicuously formed the basis of their systems. They
30 OF IDEAS IN GENERAL.
have indicated this antithesis by the terms subject Ire and
objective. According to the technical language of old
writers, a thing and its qualities are described as subject
and attributes ; and thus a man's faculties and acts are
attributes of which he is the subject. The mind is the
sulject in which ideas inhere.. Moreover, the man's
faculties and acts are employed upon external objects;
and from objects all his sensations arise. Hence the
part of a man's knowledge which belongs to his own
mind, is subjective: that which flows in upon him from
the world external to him, is objective. And as in man's
contemplation of nature, there is always some act of
thought which depends upon himself, and some matter
of thought which is independent of him, there is, in every
part of his knowledge, a subjective and an objective
element. The combination of the two elements, the
subjective or ideal, and the objective or observed, is
necessary, in order to give us any insight into the laws of
nature. But different persons, according to their mental
habits and constitution, may be inclined to dwell by
preference upon the one or the other of these two
elements. It may perhaps interest the reader to see
this difference of intellectual character illustrated in two
eminent men of genius of modern times, Gothe and
Schiller.
Gothe himself gives us the account to which I refer,
in his history of the progress of his speculations con
cerning the Metamorphosis of Plants; a mode of viewing
their structure by which he explained, in a very striking
and beautiful manner, the relations of the different parts
of a plant to each other ; as has been narrated in the
History of the Inductive Sciences. Gothe felt a delight
in the passive contemplation of nature, unmingled with
the desire of reasoning and theorizing ; a delight such as
naturally belongs to those poets who merely embody the
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 31
images which a fertile genius suggests, and do not mix
with these pictures, judgments and reflexions of their
own. Schiller, on the other hand, both by his own
strong feeling of the value of a moral purpose in poetry,
and by his adoption of a system of metaphysics in which
the subjective element was made very prominent, was
well disposed to recognize fully the authority of ideas
over external impressions.
Gothe for a time felt a degree of estrangement
towards Schiller, arising from this contrariety in their
views and characters. But on one occasion they fell
into discussion on the study of natural history; and
Gothe endeavoured to impress upon his companion his
persuasion that nature was to be considered, not as com
posed of detached and incoherent parts, but as active
and alive, and unfolding herself in each portion, in
virtue of principles which pervade the whole. Schiller
objected that no such view of the objects of natural
history had been pointed out by observation, the only
guide which the natural historians recommended; and
was disposed on this account to think the whole of their
study narrow and shallow. "Upon this," says Gothe,
" I expounded to him, in as lively a way as I could, tho
metamorphosis of plants, drawing on paper for him, as I
proceeded, a diagram to represent that general form of
a plant which shows itself in so many and so various
transformations. Schiller attended and understood; and,
accepting the explanation, he said, ' This is not observa
tion, but an idea.' I replied," adds Gothe, "with some
degree of irritation ; for the point which separated us
was most luminously marked by this expression : but I
smothered my vexation, and merely said, ' I was happy
to find that I had got ideas without knowing it ; nay,
that I saw them before my eyes.' " Gothe then goes on
to say, that he had been grieved to the very soul by
32 OF IDEAS IX GENERAL.
maxims promulgated by Schiller, that no observed fact
ever could correspond with an idea. Since he himself
loved best to wander in the domain of external observa
tion, he had been led to look with repugnance and
hostility upon anything Avhich professed to depend upon
ideas. "Yet," he observes, "it occurred to me that if
my Observation was identical with his Idea, there must
be some common ground on which we might meet."
They went on with their mutual explanations, and be
came intimate and lasting friends. "And thus," adds
the poet, " by means of that mighty and interminable
controversy between object and subject, we two concluded
an alliance which remained unbroken, and produced
much benefit to ourselves and others."
The general diagram of a plant, of which Gothe
here speaks, must have been a combination of lines and
marks expressing the relations of position and equiva
lence among the elements of vegetable forms, by which
so many of their resemblances and differences may be
explained. Such a symbol is not an Idea in that general
sense in which we propose to use the term, but is a
particular modification of the general Ideas of symmetry,
developement, and the like ; and we shall hereafter see,
according to the phraseology which we shall explain in
the next chapter, how such a diagram might express
the ideal conception of a plant.
The antithesis of subjective and objective is very
familiar in the philosophical literature of Germany and
France; nor is it uncommon in any age of our own
literature. But though efforts have recently been made
to give currency among us to this phraseology, it has
not been cordially received, and has been much com
plained of as not of obvious meaning. ]STor is the com
plaint without ground : for when we regard the mind as
the subject in which ideas inhere, it becomes for us an
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 33
object, and the antithesis vanishes. We are not so
much accustomed to use subject in this sense, as to
make it a proper contrast to object. The combination
" ideal and objective" would more readily convey to a
modern reader the opposition which is intended between
the ideas of the mind itself, and the objects which it
contemplates around it.
To the antitheses already noticed — Thoughts and
Things ; Necessary and Experiential Truths ; Deduction
and Induction ; Theory and Fact ; Ideas and Sensations ;
Reflexion and Sensation ; Subjective and Objective ; we
may add others, by which distinctions depending more
or less upon the fundamental antithesis have been de
noted. Thus we speak of the internal and external
sources of our knowledge ; of the world within and the
world without us ; of Man and Nature. Some of the
more recent metaphysical writers of Germany have
divided the universe into the Me and the Not-me (Ich
and Xicht-ich). Upon such phraseology we may observe,
that to have the fundamental antithesis of which we
speak really understood, is of the highest consequence
to philosophy, but that little appears to be gained by
expressing it in any novel manner. The most weighty
part of the philosopher's task is to analyze the operations
of the mind ; and in this task, it can aid us but little to
call it, instead of the mind, the subject, or the me.
SECT. 8. — Matter and Form.
THERE are some other ways of expressing, or rather
of illustrating, the fundamental antithesis, which I may
briefly notice. The antithesis has been at different times
presented by means of various images. One of the most
ancient of these, and one which is still very instructive,
is that which speaks of Sensations as the Matter, and
Ideas as the Form, of our knowledge : just as ivory is
VOL. i. w. P. D
34 OF IDKAS IX (JKXKUAL.
the matter, and a cube the form, of a die. This com
parison has the advantage of showing- that two elements
of an antithesis which cannot be separated in fact, may
yet be advantageously separated in our reasonings. For
Matter and Form cannot by any means be detached
from each other. All matter must have some form ; all
form must be the form of some material thing. If the
ivory be not a cube, it must have a spherical or some
other farm. And the cube, in order to be a cube, must
be of some material ; — if not of ivory, of wood, or stone,
for instance. A figure without matter is merely a geo
metrical conception ; — a modification of the idea of
space. Matter without figure is a mere abstract term ;
—a supposed union of certain sensible qualities which,
so insulated from others, cannot exist. Yet the distinc
tion of Matter and Form is real ; and, as a subject of
contemplation, clear and plain. Nor is the distinction by
any means useless. The speculations which treat of the
two subjects, Matter and Figure, are very different.
Matter is the subject of the sciences of Mechanics and
Chemistry ; Figure, of Geometry. These two classes of
Sciences have quite different sets of principles. If we
refuse to consider the Matter and the Form of bodies
separately, because we cannot exhibit Matter and Form
separately, we shut the door to all philosophy on such
subjects. In like manner, though Sensations and Ideas
are necessarily united in all our knowledge, they can be
considered as distinct; and this distinction is the basis of
all philosophy concerning knowledge.
This illustration of the relation of Ideas and Sensa
tions may enable us to estimate a doctrine which has been
put forwards at various times. In a certain school of spe
culators there has existed a disposition to derive all our
Ideas from our Sensations, the term Idea being, in this
school, used in its wider sense, so as to include all modifi-
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 35
cations and limitations of our Fundamental Ideas. The
doctrines of this school have been summarily expressed
by saying that " Every Idea is a transformed Sensation."
Now, even supposing this assertion to be exactly true,
we easily see, from what has been said, how little we
are likely to answer the ends of philosophy by putting
forward such a maxim as one of primary importance.
For we might say, in like manner, that every statue is
but a transformed block of marble, or every edifice but
a collection of transformed stones. But what would
these assertions avail us, if our object were to trace the
rules of art by which beautiful statues were formed, or
great works of architecture erected ? The question
naturally occurs, What is the nature, the principle, the
law of this Transformation ? In what faculty resides the
transforming power? What train of ideas of beauty,
and symmetry, and stability, in the mind of the statuary
or the architect, has produced those great works which
mankind look upon as among their most valuable pos
sessions ; — the Apollo of the Belvidere, the Parthenon,
the Cathedral of Cologne ? When this is what we want
to know, how are we helped by learning that the Apollo
is of Parian marble, or the Cathedral of basaltic stone ?
We must know much more than this, in order to acquire
any insight into the principles of statuary or of archi
tecture. In like manner, in order that we may make
any progress in the philosophy of knowledge, which is
our purpose, we must endeavour to learn something
further respecting ideas than that they are transformed
sensations, even if they were this.
But, in reality, the assertion that our ideas are trans
formed sensations, is erroneous as well as frivolous. For
it conveys, and is intended to convey, the opinion that
our sensations have one form which properly belongs to
them ; and that, in order to become ideas, they are con-
D 2
36 OF 1DKAS IX (JKNKKAL.
verted into some other form. But the truth is, that our
sensations, of themselves, without some act of the mind,
such as involves what we have termed an Idea, have no
form. We cannot see one object without the idea of
space ; we cannot see two without the idea of resem
blance or difference; and space and difference are not
sensations. Thus, if we are to employ the metaphor of
Matter and Form, which is implied in the expression to
which I have referred, our sensations, from their first
reception, have their Form not changed, but given by
our Ideas. Without the relations of thought which we
here term Ideas, the sensations are matter without form.
Matter without form cannot exist : and in like manner
sensations cannot become perceptions of objects, without
some formative power of the mind. By the very act of
being received as perceptions, they have a formative
power exercised upon them, the operation of which
might be expressed, by speaking of them, not as trans
formed, but simply as formed; — as invested with form,
instead of being the mere formless material of percep
tion. The word inform, according to its Latin etymo
logy, at first implied this process by which matter is
invested with form. Thus Virgil'- speaks of the thunder
bolt as informed by the hands of Brontes, and Steropes,
and Pyracmon. And Dryden introduces the word in
another place : —
Let others better mould the running mass
Of metals, or inform the breathing brass.
Even in this use of the word, the form is something
superior to the brute manner, and gives it a new signi
ficance and purpose. And hence the term is again used
* Ferrum exercebant vasto Cyclopes in Antro
Brontesque Steropesque et nudus membra Pyracmon ;
His informatum manibus, jam parte polita
Fulmen erat. — JEn. viii. 424.
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 37
to denote the effect produced by an intelligent principle
of a still higher kind:—
Hu informed
This ill-shaped body with a daring soul.
And finally even the soul itself, in its original condition,
is looked upon as matter, when viewed with reference
to education and knowledge, by which it is afterwards
moulded ; and hence these are, in our language, termed
information. If we confine ourselves to the first of
these three uses of the term, we may correct the erro
neous opinion of which we have just been speaking,
and retain the metaphor by which it is expressed, by
saying, that ideas are not transformed, but informed
sensations.
SECT. 9. — Man the Interpreter of Nature.
THKRE is another image by which writers have repre
sented the acts of thought through which knowledge is
obtained from the observation of the external world.
Nature is the Book, and Man is the Interpreter. The
facts of the external world are marks, in which man
discovers a meaning, and so reads them. Man is the
Interpreter of Nature, and Science is the right Interpre
tation. And this image also is, in many respects, instruc
tive. It exhibits to us the necessity of both elements ;—
the marks which man has to look at, and the knowledge
of the alphabet and language which he must possess and
apply before he can find any meaning in what he sees.
Moreover this image presents to us, as the ideal element,
an activity of the mind of that very kind which we wish
to point out. Indeed the illustration is rather an
example than a comparison of the composition of our
knowledge. The letters and symbols which are pre
sented to the Interpreter are really objects of sensation :
the notion of letters as signs of words, the notion of
38 OF IDEAS IN GENERAL.
connexions among words by which they have meaning,
really are among our Ideas ; — Signs and Meaning are
Ideas, supplied by the mind, and added to all that sensa
tion can disclose in any collection of visible marks. The
Sciences are not figuratively, but really, Interpretations
of Nature. But this image, whether taken as example or
comparison, may serve to show both the opposite charac
ter of the two elements of knowledge, and their neces
sary combination, in order that there may be knowledge.
This illustration may also serve to explain another
point in the conditions of human knowledge which we
shall have to notice : — namely, the very different degrees
in which, in different cases, we are conscious of the
mental act by which our sensations are converted into
knowledge. For the same difference occurs in reading
an inscription. If the inscription were entire and plain,
in a language with which we were familiar, we should
be unconscious of any mental act in reading it. We
should seem to collect its meaning by the sight alone.
But if we had to decipher an ancient inscription, of
which only imperfect marks remained, with a few entire
letters among them, we should probably make several
suppositions as to the mode of reading it, before we
found any mode which was quite successful ; and thus,
our guesses, being separate from the observed facts, and
at first not fully in agreement with them, we should be
clearly aware that the conjectured meaning, on the one
hand, and the observed marks on the other, were dis
tinct things, though these two things would become
united as elements of one act of knowledge when we
had hit upon the right conjecture.
SECT. 10. — The Fundamental Antithesis inseparable.
THE illustration just referred to, as well as other
ways of considering the subject, may help us to get over
FUNDAMENTAL ANT1THKSIS OF PHILOSOPHY. 3,9
a difficulty which at first sight appears perplexing. We
have spoken of the common opposition of Theory and
Fact as important, and as involving what we have called
the Fundamental Antithesis of Philosophy. But after
all, it may be asked. Is this distinction of Theory and
Fact really tenable? Is it not often difficult to say
whether a special part of our knowledge is a Fact or
a Theory? Is it a Fact or a Theory that the stars
revolve round the pole? Is it a Fact or a Theory that
the earth is a globe revolving on its axis? Is it a Fact
or a Theory that the earth travels in an ellipse round
the sun? Is it a Fact or a Theory that the sun attracts
the earth? Is it a Fact or a Theory that the loadstone
attracts the needle? In all these cases, probably some
persons would answer one way, and some persons the
other. There are many persons by whom the doctrine
of the globular form of the earth, the doctrine of the
earth's elliptical orbit, the doctrine of the sun's attrac
tion on the earth, would be called theories, even if they
allowed them to be true theories. But yet if each of
these propositions be true, is it not &factf And even
with regard to the simpler facts, as the motion of the
stars round the pole, although this may be a Fact to one
who has watched and measured the motions of the stars,
one who has not done this, and who has only carelessly
looked at these stars from time to time, may naturally
speak of the circles which the astronomer makes them
describe as Theories. It would seem, then, that we
cannot in such cases expect general assent, if we say,
This is a Fact and not a Theory, or, This is a Theory
and not a Fact. And the same is true in a vast range
of cases. It would seem, therefore, that we cannot rest
any reasoning upon this distinction of Theory and Fact:
and we cannot avoid asking whether there is any real
distinction in this antithesis, and if so. what it is.
40 OF IDEAS IN GENERAL.
To this I reply : the distinction between Theory
(that is, true Theory) and Fact, is this: that in Theory
the Ideas are considered as distinct from the Facts: in
Facts, though Ideas may be involved, they are not, in
our apprehension, separated from the sensations. In a
Fact, the Ideas are applied so readily and familiarly, and
incorporated with the sensations so entirely, that we
do not see them, we see through them. A person who
carefully notes the motion of a star all night, sees the
circle which it describes, as he sees the star, though
the circle is, in fact, a result of his own Ideas. A
person who has in his mind the measures of different
lines and countries on the earth's surface, and who can
put them together into one conception, finds that they
can make no figure but a globular one: to him, the
earth's globular form is a Fact, as much as the square
form of his chamber. A person to whom the grounds
of believing the earth to travel round the sun are as
familiar as the grounds for believing the movements
of the mail-coaches in this country, looks upon the
former event as a Fact, just as he looks upon the latter
events as Facts. And a person who, knowing the Fact
of the earth's annual motion, refers it distinctly to its
mechanical cause, conceives the sun's attraction as a
Fact, just as he conceives as a Fact, the action of the
wind which turns the sails of a mill. He cannot see
the force in either case ; he supplies it out of his own
Ideas. And thus, a true Theory is a Fact ; a Fact is
a familiar Theory. That which is a Fact under one
aspect, is a Theory under another. The most recondite
Theories when firmly established are Facts: the sim
plest Facts involve something of the nature of Theory.
Theory and Fact correspond, in a certain degree, with
Ideas and Sensations, as to the nature of their opposi
tion. But the Facts are Facts, so far as the Ideas have
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 41
been combined with the Sensations and absorbed in
them: the Theories are Theories, so far as the Ideas
are kept distinct from the Sensations, and so far as it is
considered still a question whether those can be made
to agree with these.
We may, as I have said, illustrate this matter by
considering man as interpreting the phenomena which
he sees. He often interprets without being aware that
he does so. Thus when we see the needle move towards
the magnet, we assert that the magnet exercises an
attractive force on the needle. But it is only by an
interpretative act of our own minds that we ascribe
this motion to attraction. That, in this case, a force is
exerted — something of the nature of the pull which we
could apply by our own volition — is our interpretation
of the phenomena; although we may be conscious of the
act of interpretation, and may then regard the attrac
tion as a Fact.
Nor is it in such cases only that we interpret phe
nomena in our own way, without being conscious of
what we do. We see a tree at a distance, and judge it
to be a chestnut or a lime ; yet this is only an inference
from the colour or form of the mass according to pre
conceived classifications of our own. Our lives are full
of such unconscious interpretations. The farmer recog
nizes a good or a bad soil ; the artist a picture of a
favourite master ; the geologist a rock of a known local
ity, as we recognize the faces and voices of our friends ;
that is, by judgments formed on what we see and hear :
but judgments in which we do not analyze the steps, or
distinguish the inference from the appearance. And in
these mixtures of observation and inference, we speak of
the judgment thus formed, as a Fact directly observed.
Even in the case in which our perceptions appear to
be most direct, and least to involve any interpretations
42 OF IDEAS IN GENERAL.
of our own, — in the simple process of seeing, — who does
not know how much we, by an act of the mind, add to
that which our senses receive ? Does any one fancy that
lie sees a solid cube ? It is easy to show that the solid
ity of the figure, the relative position of its faces and
edges to each other, are inferences of the spectator ; no
more conveyed to his conviction by the eye alone, than
they would be if he were looking at a painted represen
tation of a cube. The scene of nature is a picture with
out depth of substance, no less than the scene of art ;
and in the one case as in the other, it is the mind which,
by an act of its own, discovers that colour and shape
denote distance and solidity. Most men are unconscious
of this perpetual habit of reading the language of the
external world, and translating as they read. The
draughtsman, indeed, is compelled, for his purposes, to
return back in thought from the solid bodies which he
has inferred, to the shapes of surface which he really
sees. He knows that there is a mask of theory over the
whole face of nature, if it be tlieonj to infer more than
we see. But other men, unaware of this masquerade,
hold it to be a fact that they see cubes and spheres, spa
cious apartments and winding avenues. And these things
are facts to them, because they are unconscious of the
mental operation by which they have penetrated nature's
disguise.
And thus, we still have an intelligible distinction of
Fact and Theory, if we consider Theory as a conscious, and
Fact as an unconscious inference, from the phenomena
which are presented to our senses.
But still, Theory and Fact, Inference and Perception,
Reasoning and Observation, are antitheses in none of
which can we separate the two members by any fixed
and definite line.
Even the simplest terms by which the antithesis is
FUNDAMENTAL ANT1TIIKSIS OF PHILOSOPHY. 4.*>
expressed cannot be separated. Ideas and Sensations,
Thoughts and Things, Subject and Object, cannot in any
case be applied absolutely and exclusively. Our Sen
sations require Ideas to bind them together, namely,
Ideas of space, time, number, and the like. If not so
bound together, Sensations do not give us any appre
hension of Things or Objects. All Things, all Objects,
must exist in space and in time — must be one or many.
Now space, time, number, are not Sensations or Things.
They are something different from, and opposed to Sen
sations and Things. We have termed them Ideas. It
may be said they are Relations of Things, or of Sensa
tions. But granting this form of expression, still a
Relation is not a Thing or a Sensation ; and therefore
we must still have another and opposite element, along
with our Sensations. And yet, though we have thus
these two elements in every act of perception, we cannot
designate any portion of the act as absolutely and exclu
sively belonging to one of the elements. Perception
involves Sensation, along with Ideas of time, space, and
the like ; or, if any one prefers the expression, we may
say, Perception involves Sensations along with the ap
prehension of Relations. Perception is Sensation, along
with such Ideas as make Sensation into an apprehension
of Things or Objects.
And as Perception of Objects implies Ideas, — as Ob
servation implies Reasoning; — so, on the other hand.
Ideas cannot exist where Sensation has not been ; Rea
soning cannot go on when there has not been previous
Observation. This is evident from the necessary order
of developement of the human faculties. Sensation
necessarily exists from the first moments of our exist
ence, and is constantly at work. Observation begins
before we can suppose the existence of any Reasoning
which is not involved in Observation. Hence, at what-
44 OF IDEAS IN GENERAL.
ever period we consider our Ideas, we must consider
them as having been already engaged in connecting our
Sensations, and as having been modified by this employ
ment. By being so employed, our Ideas are unfolded
and defined ; and such developement and definition can
not be separated from the Ideas themselves. We cannot
conceive space, without boundaries or forms ; now Forms
involve Sensations. We cannot conceive time, without
events which mark the course of time ; but events involve
Sensations. We cannot conceive number, without con
ceiving things which are numbered ; and Things imply
sensations. And the forms, things, events, which are
thus implied in our Ideas, having been the objects of
Sensation constantly in every part of our life, have
modified, unfolded, and fixed our Ideas, to an extent
which we cannot estimate, but which we must suppose
to be essential to the processes which at present go on
in our minds. We cannot say that Objects create Ideas ;
for to perceive Objects we must already have Ideas.
But we may say, that Objects and the constant Perception
of Objects have so far modified our Ideas, that we cannot,
oven in thought, separate our Ideas from the perception
of Objects.
We cannot say of any Ideas, as of the Idea of space,
or time, or number, that they are absolutely and exclu
sively Ideas. We cannot conceive what space, or time,
or number, would be in our minds, if we had never per
ceived any Thing or Things in space or time. We can
not conceive ourselves in such a condition as never to have
perceived any Thing or Things in space or time. But, on
the other hand, just as little can we conceive ourselves
becoming acquainted with space and time or numbers
as objects of Sensation. We cannot reason without
having the operations of our minds affected by previous
Sensations ; but we cannot conceive Reasoning to be
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 45
merely a series of Sensations. In order to be used in
Reasoning, Sensation must become Observation ; and, as
we have seen, Observation already involves Reasoning.
In order to be connected by our Ideas, Sensations must
be Things or Objects, and Things or Objects already in
clude Ideas. And thus, none of the terms by which the
fundamental antithesis is expressed can be absolutely
and exclusively applied.
I will make a remark suggested by the views which
have thus been presented. Since, as we have just seen,
none of the terms which express the fundamental anti
thesis can be applied absolutely and exclusively, the
absolute application of the antithesis in any particular
case can never be a conclusive or immoveable principle.
This remark is the more necessary to be borne in mind, as
the terms of this antithesis are often used in a vehement
and peremptory manner. Thus we are often told that
such a thing is a Fact; A FACT and not a Theory, with all
the emphasis which, in speaking or writing, tone or italics
or capitals can give. We see from what has been said,
that when this is urged, before we can estimate the
truth, or the value of the assertion, we must ask to
whom is it a Fact? what habits of thought, what pre
vious information, what Ideas does it imply, to conceive
the Fact as a Fact ? Does not the apprehension of the
Fact imply assumptions which may with equal justice
be called Theory, and which are perhaps false Theory?
in which case, the Fact is no Fact. Did not the an
cients assert it as a Fact, that the earth stood still,
and the stars moved ? and can any Fact have stronger
apparent evidence to justify persons in asserting it em
phatically than this had ?
These remarks are by no means urged in order to
shew that no Fact can be certainly known to be true ;
but onlv, to shew that no Fact can be certainlv shown
4(i OF IDEAS IN GENERAL.
to be a Fact, merely by calling it a Fact, however
emphatically. There is by no means any ground of
general skepticism with regard to truth, involved in
the doctrine of the necessary combination of two ele
ments in all our knowledge. On the contrary, Ideas
are requisite to the essence, and Things to the reality
of our knowledge in every case. The proportions of
Geometry and Arithmetic are examples of knowledge
respecting our Ideas of space and number, with regard
to which there is no room for doubt. The doctrines of
Astronomy are examples of truths not less certain
respecting the Facts of the external world.
SECT. 11. — Successive Generalization.
IN the preceding pages we have been led to the doctrine,
that though, in the Antithesis of Theory and Fact, there
is involved an essential opposition ; namely the opposition
of the thoughts within us and the phenomena without
us ; yet that we cannot distinguish and define the mem
bers of this antithesis separately. Theories become
Facts, by becoming certain and familiar : and thus, as
our knowledge becomes more sure and more extensive,
we are constantly transferring to the class of facts,
opinions which were at first regarded as theories.
Now we have further to remark, that in the progress
of human knowledge respecting any branch of specula
tion, there may be several such steps in succession, each
depending upon and including the preceding. The
theoretical views which one generation of discoverers
establishes, become the facts from which the next gene
ration advances to new theories. As men rise from the
particular to the general, so, in the same manner, they
rise from what is general to what is more general. Each
induction supplies the materials of fresh inductions ;
each generalization, with all that it embraces in its circle.
FUN'DAMKNTAL AXTITHKSIS OF PUILOHIPFI Y. 47
may be found to be but one of many circles, compre
hended within the circuit of some wider generalization.
This remark has already been made, and illustrated,
in the History of the Inductive Sciences"' ; and, in truth,
the whole of the history of science is full of suggestions
and exemplifications of this course of things. It may be
convenient, however, to select a few instances which may
further explain and confirm this view of the progress of
scientific knowledge.
The most conspicuous instance of this succession is
to be found in that science which has been progressive
from the beginning of the world to our own times, and
which exhibits by far the richest collection of successive
discoveries : I mean Astronomy. It is easy to see that
each of these successive discoveries depended on those
antecedently made, and that in each, the truths which
were the highest point of the knowledge of one age
were the fundamental basis of the efforts of the age
which came next. Thus we find, in the days of Greek
discovery, Hipparchus and Ptolemy combining and ex
plaining the particular facts of the motion of the sun,
moon, and planets, by means of the theory of epicycles
and eccentrics ; — a highly important step, which gave
an intelligible connexion and rule to the motions of each
of these luminaries. When these cycles and epicycles,
thus truly representing the apparent motions of the
heavenly bodies, had accumulated to an inconvenient
amount, by the discovery of many inequalities in the
observed motions, Copernicus showed that their effects
might all be more simply included, by making the sun
the center of motion of the planets, instead of the earth.
But in this new view, he still retained the epicycles and
eccentrics which governed the motion of each body.
Tycho Brahe's observations, and Kepler's calculations,
f Uixf. Inductive Sciences, B. vn c. ii. Sort. "».
48 OF IDEAS JN GENERAL.
showed that, besides the vast number of facts which the
epicyclical theory could account for, there were some
which it would not exactly include, and Kepler was led
to the persuasion that the planets move in ellipses.
But this view of motion was at first conceived by Kepler
as a modification of the conception of epicycles. On one
occasion he blames himself for not sooner seeing that
such a modification was possible. " What an absurdity
on my part !" he cries"'- ; " as if libration in the diameter
of the epicycle might not come to the same thing as
motion in the ellipse." But again; Kepler's laws of the
elliptical motion of the planets were established ; and
these laws immediately became the facts on which the
mathematicians had to found their mechanical theories.
From these facts, Newton, as we have related, proved
that the central force of the sun retains the planets in
their orbits, according to the law of the inverse square
of the distance. The same law was shown to prevail in
the gravitation of the earth. It was shown, too, by in
duction from the motions of Jupiter and Saturn, that
the planets attract each other ; by calculations from the
figure of the earth, that the parts of the earth attract
each other ; and, by considering the course of the tides,
that the sun and moon attract the waters of the ocean.
And all these curious discoveries being established as
facts, the subject was ready for another step of gene
ralization. By an unparalleled rapidity in the progress
of discovery in this case, not only were all the inductions
which we have first mentioned made by one individual,
but the new advance, the higher flight, the closing vic
tory, fell to the lot of the same extraordinary person.
The attraction of the sun upon the planets, of the
moon upon the earth, of the planets on each other, of the
parts of the earth on themselves, of the sun and moon
* Hist. Inductive Sciences, B. v. c. iv. Sect. 3.
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 49
upon the ocean; — all these truths, each of itself a great
discovery, were included by Newton in the higher gene
ralization, of the universal gravitation of matter, by
which each particle is drawn to each other according to
the law of the inverse square : and thus this long ad
vance from discovery to discovery, from truths to truths,
each justly admired when new, and then rightly used as
old, was closed in a worthy and consistent manner, by
a truth which is the most worthy admiration, because it
includes all the researches of preceding ages of Astro
nomy.
We may take another example of a succession of this
kind from the history of a science, which, though it has
made wonderful advances, has not yet reached its goal,
as physical astronomy appears to have done, but seems to
have before it a long prospect of future progress. I now
refer to Chemistry, in which I shall try to point out how
the preceding discoveries afforded the materials of the
succeeding; although this subordination and connexion
is, in this case, less familiar to men's minds than in Astro
nomy, and is, perhaps, more difficult to present in a clear
and definite shape. Sylvius saw. in the facts which
occur, when an acid and an alkali are brought together,
the evidence that they neutralize each other. But cases
of neutralization, and acidification, and many other ef
fects of mixture of the ingredients of bodies, being thus
viewed as facts, had an aspect of unity and law given
them by Geoffroy and Bergman""", who introduced the con
ception of the Chemical Affinity or Elective Attraction,
by which certain elements select other elements, as if by
preference. That combustion, whether a chemical union
or a chemical separation of ingredients, is of the same
nature with acidification, was the doctrine of Beccher
* Hixt. Inductive Sciences, B. xiv. c. iii.
VOL. I. W. P. E
•30 OF IDEAS IN GENERAL.
and Stahl, and was soon established as a truth which
must form a part of every succeeding physical theory.
That the rules of affinity and chemical composition may
include gaseous elements, was established by Black and
Cavendish. And all these truths, thus brought to light
by chemical discoverers, — affinity, the identity of acidifi
cation and combustion, the importance of gaseous ele
ments, — along with all the facts respecting the weight
of ingredients and compounds which the balance dis
closed, — were taken up, connected, and included as
particulars in the oxygen theory of Lavoisier. Again,
the results of this theory, and the quantity of the several
ingredients which entered into each compound — (such
results, for the most part, being now no longer mere
theoretical speculations, but recognized facts) — were the
particulars from which Dalton derived that wide law of
chemical combination which we term the Atomic Theory.
And this law, soon generally accepted among chemists,
is already in its turn become one of the facts included
in Faraday's Theory of the identity of Chemical Affinity
and Electric Attraction.
It is unnecessary to give further exemplifications of
this constant ascent from one step to a higher ; — this
perpetual conversion of true theories into the materials
of other and wider theories. It will hereafter be our
business to exhibit, in a more full and formal manner,
the mode in which this principle determines the whole
scheme and structure of all the most exact sciences.
And thus, beginning with the facts of sense, we gradually
climb to the highest forms of human knowledge, and
obtain from experience and observation a vast collection
of the most wide and elevated truths.
There are, however, truths of a very different kind, to
which we must turn our attention, in order to pursue our
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 51
researches respecting the nature and grounds of our
knowledge. But before we do this, we must notice one
more feature in that progress of science which we have
already in part described.
CHAPTER III.
OF TECHNICAL TERMS.
1 . IT has already been stated that we gather knowledge
from the external world, when we are able to apply, to
the facts which we observe, some ideal conception, which
gives unity and connexion to multiplied and separate
perceptions. We have also shown that our conceptions,
thus verified by facts, may themselves be united and con
nected by a new bond of the same nature ; and that man
may thus have to pursue his way from truth to truth
through a long progression of discoveries, each resting
on the preceding, and rising above it.
Each of these steps, in succession, is recorded, fixed,
and made available, by some peculiar form of words ;
and such words, thus rendered precise in their meaning,
and appropriated to the service of science, we may call
Technical Terms. It is in a great measure by inventing
such Terms that men not only best express the discoveries
they have made, but also enable their followers to become
so familiar with these discoveries, and to possess them
so thoroughly, that they can readily use them in ad
vancing to ulterior generalizations.
Most of our ideal conceptions are described by exact
and constant words or phrases, such as those of which we
here speak. We have already had occasion to employ
many of these. Thus we have had instances of technical
Terms expressing geometrical conceptions, as Ellipsis,
E2
OZ OF IDEAS IN GENERAL.
Radius Vector, Axis, Plane, the Proportion of the In
verse Square, and the like. Other Terms have described
mechanical conceptions, as Accelerating Force and
Attraction. Again, chemistry exhibits (as do all sciences)
a series of Terms which mark the steps of our progress.
The views of the first real founders of the science are
recorded by the Terms which are still in use, Neutral
Salts, Affinity, and the like. The establishment of Dai-
ton's theory has produced the use of the word Atom in
a peculiar sense, or of some other word, as Proportion,
in a sense equally technical. And Mr. Faraday has
found it necessary, in order to expound his electro-chemi
cal theory, to introduce such terms as Anode and Cathode,
Anion and CatMon.
2. I need not adduce any further examples, for my
object at present is only to point out the use and influence
of such language : its rules and principles I shall here
after try, in some measure, to fix. But what we have
here to remark is, the extraordinary degree in which the
progress of science is facilitated, by thus investing each
new discovery with a compendious and steady form of
expression. These terms soon become part of the cur
rent language of all who take an interest in speculation.
However strange they may sound at first, they soon grow
familiar in our ears, and are used without any effort, or
any recollection of the difficulty they once involved. They
become as common as the phrases which express our
most frequent feelings and interests, while yet they have
incomparably more precision than belongs to any terms
which express feelings; and they carry with them, in
their import, the results of deep and laborious trains of
research. They convey the mental treasures of one
period to the generations that follow ; and laden with
this, their precious freight, they sail safely across gulfs
of time in which empires have suffered shipwreck, and
OF TECHNICAL TERMS. 53
the languages of common life have sunk into oblivion.
We have still in constant circulation among us the Terms
which belong to the geometry, the astronomy, the
zoology, the medicine of the Greeks, and the algebra
and chemistry of the Arabians. And we can in an in
stant, by means of a few words, call to our own recollec
tion, or convey to the apprehension of another person,
phenomena and relations of phenomena in optics, mine
ralogy, chemistry, which are so complex and abstruse,
that it might seem to require the utmost subtlety of the
human mind to grasp them, even if that were made the
sole object of its efforts. By this remarkable effect of
Technical Language, we have the results of all the
labours of past times not only always accessible, but so
prepared that we may (provided we are careful in the
use of our instrument) employ what is really useful and
efficacious for the purpose of further success, without
being in any way impeded or perplexed by the length
and weight of the chain of past connexions which we
drag along with us.
By such means, — by the use of the Inductive Process,
and by the aid of Technical Terms, — man has been con
stantly advancing in the path of scientific truth. In a
succeeding part of this work we shall endeavour to trace
the general rules of this advance, and to lay down the
maxims by which it may be most successfully guided
and forwarded. But in order that we may do this to
the best advantage, we must pursue still further the
analysis of knowledge into its elements ; and this will be
our employment in the first part of the work.
54
CHAPTER IV.
OF NECESSARY TRUTHS.
1 . EVERY advance in human knowledge consists, as
we have seen, in adapting new ideal conceptions to ascer
tained facts, and thus in superinducing the Form upon
the Matter, the active upon the passive processes of our
minds. Every such step introduces into our knowledge
an additional portion of the ideal element, and of those
relations which flow from the nature of Ideas. It is,
therefore, important for our purpose to examine more
closely this element, and to learn what the relations are
which may thus come to form part of our knowledge.
An inquiry into those Ideas which form the foundations
of our sciences ; — into the reality, independence, extent,
and principal heads of the knowledge which we thus ac
quire ; — is a task on which we must now enter, and
which will employ us for several of the succeeding Books.
In this inquiry our object will be to pass in review all
the most important Fundamental Ideas which our
sciences involve ; and to prove more distinctly in refer
ence to each, what we have already asserted with regard
to all, that there are everywhere involved in our know
ledge acts of the mind as well as impressions of sense ;
and that our knowledge derives, from these acts, a gene
rality, certainty, and evidence which the senses could in
no degree have supplied. But before I proceed to do
this in particular cases, I will give some account of the
argument in its general form.
We have already considered the separation of our
knowledge into its two elements, — Impressions of Sense
and Ideas, — as evidently indicated by this ; that all know
ledge possesses characters which neither of these ele
ments alone could bestow. Without our ideas, our sen
sations could have no connexion ; without external
OF NECESSARY TRUTHS. 55
impressions, our ideas would have no reality ; and thus
both ingredients of our knowledge must exist.
2. There is another mode in which the distinction of
the two elements of knowledge appears, as I have already
said : (C. i. Sect. 2.) namely in the distinction of neces
sary and contingent or experiential truths. For of these
two classes of truths, the difference arises from this ;—
that the one class derives its nature from the one, and
the other from the other, of the two elements of know
ledge. I have already stated briefly the difference of
these two kinds of truths : — namely, that the former are
truths which, we see, must be true : — the latter are true,
but so far as we can see, might be otherwise. The former
are true necessarily and universally : the latter are learnt
from experience and limited by experience. Now with
regard to the former kind of truths, I wish to show that
the universality and necessity which distinguish them
can by no means be derived from experience ; that these
characters do in reality flow from the ideas which these
truths involve ; and that when the necessity of the truth
is exhibited in the way of logical demonstration, it is
found to depend upon certain fundamental principles,
(Definitions and Axioms,) which may thus be considered
as expressing, in some measure, the essential characters
of our ideas. These fundamental principles I shall after
wards proceed to discuss and to exhibit in each of the
principal departments of science.
I shall begin by considering Necessary Truths more
fully than I have yet done. As I have already said,
necessary truths are those in which we not only learn
that the proposition is true, but see that it must be true ;
in which the negation of the truth is not only false, but
impossible; in which we cannot, even by an effort of
imagination, or in a supposition, conceive the reverse of
that which is asserted.
50 OF IDEAS IN GENERAL.
3. That there are such truths cannot he doubted.
We may take, for example, all relations of number.
Three and Two added together make Five. We cannot
conceive it to be otherwise. We cannot, by any freak
of thought, imagine Three and Two to make Seven.
It may be said that this assertion merely expresses
what we mean by our words ; that it is a matter of defi
nition ; that the proposition is an identical one.
But this is by no means so. The definition of Five
is not Three and Two, but Four and One. How does it
appear that Three and Two is the same number as Four
and One ? It is evident that it is so ; but why is it evi
dent ? — not because the proposition is identical ; for if
that were the reason, all numerical propositions must be
evident for the same reason. If it be a matter of defi
nition that 3 and 2 make 5, it must be a matter of defi
nition that 39 and 27 make G6. But who will say that
the definition of 66 is 39 and 27 ? Yet the magnitude
of the numbers can make no difference in the ground of
the truth. How do we know that the product of 13 and
17 is 4 less than the product of 15 and 15? We see
that it is so, if we perform certain operations by the rules
of arithmetic ; but how do we know the truth of the
rules of arithmetic? If we divide 123375 by 987 ac
cording to the process taught us at school, how are we
assured that the result is correct, and that the number
125 thus obtained is really the number of times one
number is contained in the other?
The correctness of the rule, it may be replied, can be
rigorously demonstrated. It can be shewn that the pro
cess must inevitably give the true quotient.
Certainly this can be shown to be the case. And
precisely because it can be shown that the result must be
true, we have here an example of a necessary truth ; and
this truth, it appears, is not therefore necessary because it
OF NECESSARY TRUTHS. 57
is itself evidently identical, however it may be possible to
prove it by reducing it to evidently identical propositions.
And the same is the case with all other numerical propo
sitions ; for, as we have said, the nature of all of them is
the same.
Here, then, we have instances of truths which are
not only true, but demonstrably and necessarily true.
Now such truths are, in this respect at least, altogether
different from truths, which, however certain they may
be, are learnt to be so only by the evidence of observa
tion, interpreted, as observation must be interpreted, by
our own mental faculties. There is no difficulty in find
ing examples of these merely observed truths. We find
that sugar dissolves in water, and forms a transparent
fluid, but no one will say that we can see any reason
beforehand why the result must be so. We find that all
animals which chew the cud have also the divided hoof;
but could any one have predicted that this would be
universally the case ? or supposing the truth of the rule
to be known, can any one say that he cannot conceive
the facts as occurring otherwise ? Water expands when
it crystallizes, some other substances contract in the same
circumstances ; but can any one know that this will be
so otherwise than by observation ? We have here propo
sitions rigorously true, (we will assume,) but can any
one say they are necessarily true ? These, and the great
mass of the doctrines established by induction, are actual,
but so far as we can see, accidental laws ; results deter
mined by some unknown selection, not demonstrable
consequences of the essence of things, inevitable and
perceived to be inevitable. According to the phrase
ology which has been frequently used by philosophical
writers, they arc contingent, not necessary truths.
It is requisite to insist upon this opposition, because
no insight can be obtained into the true nature of
58 OF IDEAS IN GENERAL.
knowledge, and the mode of arriving at it, by any one
who does not clearly appreciate the distinction. The
separation of truths which are learnt by observation, and
truths which can be seen to be true by a pure act of
thought, is one of the first and most essential steps in
our examination of the nature of truth, and the mode of
its discovery. If any one does not clearly comprehend
this distinction of necessary and contingent truths, he
will not be able to go along with us in our researches
into the foundations of human knowledge ; nor, indeed,
to pursue with success any speculation on the subject.
But, in fact, this distinction is one that can hardly fail
to be at once understood. It is insisted upon by almost
all the best modern, as well as ancient, metaphysicians"''',
as of primary importance. And if any person does not
fully apprehend, at first, the different kinds of truth thus
pointed out, let him study, to some extent, those sciences
which have necessary truth for their subject, as geometry,
or the properties of numbers, so as to obtain a familiar
acquaintance with such truth ; and he will then hardly
fail to see how different the evidence of the propositions
which occur in these sciences, is from the evidence of
the facts which are merely learnt from experience.
That the year goes through its course in 365 days, can
only be known by observation of the sun or stars : that
365 days is 52 weeks and a day, it requires no expe
rience, but only a little thought to perceive. That bees
build their cells in the form of hexagons, we cannot
know without looking at them ; that regular hexagons
may be arranged so as to fill space, may be proved with
the utmost rigour, even if there were not in existence
such a thing as a material hexagon.
4. As I have already said, one mode in which we
may express the difference of necessary truths and truths
* Aristotle, Dr. Wliatdy, Dugald Stewart, &c.
OF NECESSARY TRUTHS. 51)
of experience, is, that necessary truths are those of which
we cannot distinctly conceive the contrary. We can
very readily conceive the contrary of experiential truths.
We can conceive the stars moving about the pole or
across the sky in any kind of curves with any velocities ;
we can conceive the moon always appearing during the
whole month as a luminous disk, as she might do if her
light were inherent and not borrowed. But we cannot
conceive one of the parallelograms on the same base and
between the same parallels larger than the other ; for
we find that, if we attempt to do this, when we separate
the parallelograms into parts, we have to conceive one
triangle larger than another, both having all their parts
equal ; which we cannot conceive at all, if we conceive
the triangles distinctly. We make this impossibility
more clear by conceiving the triangles to be placed so
that two sides of the one coincide with two sides of the
other ; and it is then seen, that in order to conceive the
triangles unequal, we must conceive the two bases which
have the same extremities both ways, to be different
lines, though both straight lines. This it is impossible
to conceive : we assent to the impossibility as an axiom,
when it is expressed by saying, that two straight lines
cannot inclose a space ; and thus we cannot distinctly
conceive the contrary of the proposition just mentioned
respecting parallelograms.
But it is necessary, in applying this distinction, to
bear in mind the terms of it ; — that we cannot distinct/)/
conceive the contrary of a necessary truth. For in a
certain loose, indistinct way, persons conceive the con
trary of necessary geometrical truths, when they erro
neously conceive false propositions to be true. Thus,
Hobbes erroneously held that he had discovered a means
of geometrically doubling the cube, as it is called, that
is, finding two mean proportionals between two given
GO OF IDEAS IN GENERAL.
lines ; a problem which cannot be solved by plane
geometry. Hobbes not only proposed a construction for
this purpose, but obstinately maintained that it was
right, when it had been proved to be wrong. But then,
the discussion showed how indistinct the geometrical
conceptions of Hobbes were ; for when his critics had
proved that one of the lines in his diagram would not
meet the other in the point which his reasoning sup
posed, but in another point near to it ; he maintained, in
reply, that one of these points was large enough to
include the other, so that they might be considered as
the same point. Such a mode of conceiving the oppo
site of a geometrical truth, forms no exception to the
assertion, that this opposite cannot be distinctly con
ceived.
In like manner, the indistinct conceptions of children
and of rude savages do not invalidate the distinction cf
necessary and experiential truths. Children and savages
make mistakes even with regard to numbers ; and might
easily happen to assert that 27 and 38 are equal to G3
or G4. But such mistakes cannot make arithmetical
truths cease to be necessary truths. When any person
conceives these numbers and their addition distinctly, by
resolving them into parts, or in any other way, he sees
that their sum is necessarily G5. If, on the ground of
the possibility of children and savages conceiving some
thing different, it be held that this is not a necessary
truth, it must be held on the same ground, that it is not
a necessary truth that 7 and 4 are equal to 11 ; for
children and savages might be found so unfamiliar with
numbers as not to reject the assertion that 7 and 4 are
10, or even that 4 and 3 are G, or 8. But I suppose
that no persons would on such grounds hold that these
arithmetical truths arc truths known only by experi
ence.
OF NECESSARY TRUTHS. 61
5. I have taken examples of necessary truths from
the properties of number and space; but such truths exist
no less in other subjects, although the discipline of
thought which is requisite to perceive them distinctly,
may not be so usual among men with regard to the
sciences of mechanics and hydrostatics, as it is with
regard to the sciences of geometry and arithmetic. Yet
every one may perceive that there are such truths in
mechanics. If I press the table with my hand, the
table presses my hand with an equal force : here is a
self-evident and necessary truth. In any machine,
constructed in whatever manner to increase the force
which I can exert, it is certain that what I gain in force
I must lose in the velocity which I communicate. This
is not a contingent truth, borrowed from and limited by
observation ; for a man of sound mechanical views applies
it with like confidence, however novel be the construc
tion of the machine. When I come to speak of the ideas
which are involved in our mechanical knowledge, I
may, perhaps, be able to bring more clearly into view
the necessary truth of general propositions on such
subjects. That reaction is equal and opposite to action,
is as necessarily true as that two straight lines cannot
inclose a space ; it is as impossible theoretically to make
a perpetual motion by mere mechanism as to make the
diagonal of a square commensurable with the side.
6. Necessary truths must be universal truths. If any
property belong to a right-angled triangle necessarily, it
must belong to all right-angled triangles. And it shall
be proved in the following Chapter, that truths possess
ing these two characters, of Necessity and Universality,
cannot possibly be the mere results of experience.
02
CHAPTER V.
OF EXPERIENCE.
1. I HERE employ the term Experience in a more defi
nite and limited sense than that which it possesses in
common usage ; for I restrict it to matters belonging to
the domain of science. In such cases, the knowledge
which we acquire, by means of experience, is of a clear
and precise nature ; and the passions and feelings and
interests, which make the lessons of experience in prac
tical matters so difficult to read aright, no longer disturb
and confuse us. We may, therefore, hope, by attending
to such cases, to learn what efficacy experience really
has, in the discovery of truth.
That from experience (including intentional expe
rience, or observation,} we obtain much knowledge which
is highly important, and which could not be procured
from any other source, is abundantly clear. We have
already taken several examples of such knowledge.
We know by experience that animals which ruminate
are cloven-hoofed; and we know this in no other man
ner. We know, in like manner, that all the planets and
their satellites revolve round the sun from west to east.
It has been found by experience that all meteoric stones
contain chrome. Many similar portions of our know
ledge might be mentioned.
Now what we have here to remark is this ; — that in
no case can experience prove a proposition to be neces
sarily or universally true. However many instances we
may have observed of the truth of a proposition, yet if it be
known merely by observation, there is nothing to assure
us that the next case shall not be an exception to the rule.
If it be strictly true that every ruminant animal yet
known has cloven hoofs, we still cannot be sure that
OF EXPERIENCE. G3
some creature will not hereafter be discovered which has
the first of these attributes without having the other.
When the planets and their satellites, as far as Saturn, had
been all found to move round the sun in one direction,
it was still possible that there might be other such bodies
not obeying this rule ; and, accordingly, when the satel
lites of Uranus were detected, they appeared to offer an
exception of this kind. Even in the mathematical sciences,
we have examples of such rules suggested by experience,
and also of their precariousness. However far they may
have been tested, we cannot depend upon their correct
ness, except we see some reason for the rule. For
instance, various rules have been given, for the purpose
of pointing out prime numbers; that is, those which can
not be divided by any other number. We may try, as
an example of such a rule, this one — any odd power of
the number two, diminished by one. Thus the third
power of two, diminished by one, is seven; the fifth
power, diminished by one, is thirty-one ; the seventh
power so diminished is one hundred and twenty-seven.
All these are prime numbers : and we might be led to
suppose that the rule is universal. But the next ex
ample shows us the fallaciousness of such a belief. The
ninth power of two, diminished by one, is five hundred
and eleven, which is not a prime, being divisible by seven.
Experience must always consist of a limited number
of observations. And, however numerous these may be,
they can show nothing with regard to the infinite
number of cases in which the experiment has not been
made. Experience being thus unable to prove a fact
to be universal, is, as will readily be seen, still more
incapable of proving a truth to be necessary. Expe
rience cannot, indeed, offer the smallest ground for the
necessity of a proposition. She can observe and record
what has happened ; but she cannot find, in any case, or
04 OF IDEAS IN GENERAL.
in any accumulation of cases, any reason for what mutt
happen. She may see objects side by side ; but she
cannot see a reason why they must ever be side by side.
She finds certain events to occur in succession ; but the
succession supplies, in its occurrence, no reason for its
recurrence. She contemplates external objects ; but she
cannot detect any internal bond, which indissolubly
connects the future with the past, the possible with the
real. To learn a proposition by experience, and to see
it to be necessarily true, are two altogether different pro
cesses of thought.
2. But it may be said, that we do learn by means
of observation and experience many universal truths ;
indeed, all the general truths of which science consists.
Is not the doctrine of universal gravitation learnt by
experience ? Are not the laws of motion, the properties
of light, the general principles of chemistry, so learnt ?
How, with these examples before us, can we say that
experience teaches no universal truths?
To this we reply, that these truths can only be
known to be general, not universal, if they depend upon
experience alone. Experience cannot bestow that uni
versality which she herself cannot have, and that necessity
of which she has no comprehension. If these doctrines
are universally true, this universality flows from the ideas
which we apply to our experience, and which are, as we
have seen, the real sources of necessary truth. How far
these ideas can communicate their universality and
necessity to the results of experience, it will hereafter
be our business to consider. It will then appear, that
when the mind collects from observation truths of a wide
and comprehensive kind, which approach to the sim
plicity and universality of the truths of pure science ;
she gives them this character by throwing upon them
the light of her own Fundamental Ideas.
OF EXPERIENCE. (55
But the truths which we discover by observation of
the external world, even when most, strikingly simple
and universal, are not necessary truths. Is the doctrine
of universal gravitation necessarily true ? It was doubted
by Clairaut (so far as it refers to the moon), when the
progression of the apogee in fact appeared to be twice
as great as the theory admitted. It has been doubted,
even more recently, with respect to the planets, their
mutual perturbations appearing to indicate a deviation
from the law. It is doubted still, by some persons, with
respect to the double stars. But suppose all these
doubts to be banished, and the law to be universal ; is it
then proved to be necessary ? Manifestly not : the very
existence of these doubts proves that it is not so. For
the doubts were dissipated by reference to observation
and calculation, not by reasoning on the nature of the
law. Clairaut's difficulty was removed by a more exact
calculation of the effect of the sun's force on the motion
of the apogee. The suggestion of Bessel, that the in
tensity of gravitation might be different for different
planets, was found to be unnecessary, when Professor
Airy gave a more accurate determination of the mass of
Jupiter. And the question whether the extension of the
law of the inverse square to the double stars be true,
(one of the v most remarkable questions now before the
scientific world,) must be answered, not by any specula
tions concerning what the laws of attraction must neces
sarily be, but by carefully determining the actual laws
of the motion of these curious objects, by means of the
observations such as those which Sir John Hcrschel has
collected for that purpose, by his unexampled survey of
both hemispheres of the sky. And since the extent of
this truth is thus to be determined by reference to ob
served facts, it is clear that no mere accumulation of
VOL. i. w. r. F
06 OF IDEAS IN GENERAL.
them can make its universality certain, or its necessity
apparent.
Thus no knowledge of the necessity of any truths
can result from the observation of what really happens.
This being clearly understood, we are led to an import
ant inquiry.
The characters of universality and necessity in the
truths which form part of our knowledge, can never
be derived from experience, by which so large a part
of our knowledge is obtained. But since, as we have
seen, we really do possess a large body of truths which
are necessary, and because necessary, therefore universal,
the question still recurs, from what source these charac
ters of universality and necessity are derived.
The answer to this question we will attempt to give
in the next chapter.
CHAPTER VI.
OF THE GROUNDS OF NECESSARY TRUTHS.
1 . To the question just stated, I reply, that the neces
sity and universality of the truths which form a part of
our knowledge, are derived from the Fundamental Ideas
which those truths involve. These ideas entirely shape
and circumscribe our knowledge ; they regulate the ac
tive operations of our minds, without which our passive
sensations do not become knowledge. They govern
these operations, according to rules which are not only
fixed and permanent, but which may be expressed in
plain and definite terms; and these rules, when thus
expressed, may be made the basis of demonstrations by
which the necessary relations imparted to our know
ledge by our Ideas may be traced to their consequences
in the most remote ramifications of scientific truth.
GROUNDS OF NECESSARY TRUTHS. G7
These enunciations of the necessary and evident con
ditions imposed upon our knowledge by the Fundamental
Ideas which it involves, are termed Axioms. Thus the
Axioms of Geometry express the necessary conditions
which result from the Idea of Space ; the Axioms of
Mechanics express the necessary conditions which flow
from the Ideas of Force and Motion ; and so on.
2. It will be the office of several of the succeeding
Books of this work to establish and illustrate in detail
what I have thus stated in general terms. I shall there
pass in review many of the most important fundamental
ideas on which the existing body of our science depends ;
and I shall endeavour to show, for each such idea in
succession, that knowledge involves an active as well as
a passive element ; that it is not possible without an act
of the mind, regulated by certain laws. I shall further
attempt to enumerate some of the principal fundamental
relations which each idea thus introduces into our
thoughts, and to express them by means of definitions
and axioms, and other suitable forms.
I will only add a remark or two to illustrate further
this view of the ideal grounds of our knowledge.
3. To persons familiar with any of the demonstrative
sciences, it will be apparent that if we state all the
Definitions and Axioms which are employed in the
demonstrations, we state the whole basis on which those
reasonings rest. For the whole process of demonstrative
or deductive reasoning in any science, (as in geometry,
for instance,) consists entirely in combining some of these
first principles so as to obtain the simplest propositions
of. the science; then combining these so as to obtain
other propositions of greater complexity; and so on, till
we advance to the most recondite demonstrable truths ;
these last, however, intricate and unexpected, still in
volving no principles except the original definitions and
F 2
68 OF IDEAS IN GENERAL.
axioms. Thus, by combining the Definition of a triangle,
and the Definitions of equal lines and equal angles,
namely, that they are such as when applied to each
other, coincide, with the Axiom respecting straight lines
(that two such lines cannot inclose a space,) we demon
strate the equality of triangles, under certain assumed
conditions. Again, by combining this result with the
Definition of parallelograms, and with the Axiom that if
cquals be taken from equals the wholes- are equal, we
prove the equality of parallelograms between the same
parallels and upon the same base. From this proposi
tion, again, we prove the equality of the square on the
hypotenuse of a triangle to the squares on the two sides
containing the right angle. But in all this there is
nothing contained which is not rigorously the result of
our geometrical Definitions and Axioms. All the rest
of our treatises of geometry consists only of terms and
phrases of reasoning, the object of which is to connect
those first principles, and to exhibit the effects of their
combination in the shape of demonstration.
4. This combination of first principles takes place
according to the forms and rules of Logic. All the
steps of the demonstration may be stated in the shape in
which logicians are accustomed to exhibit processes of
reasoning in order to show their conclusiveness, that is,
in Syllogisms. Thus our geometrical reasonings might
be resolved into such steps as the following : —
All straight lines drawn from the centre of a circle
to its circumference are equal :
But the straight lines AB, AC, are drawn from the
centre of a circle to its circumference : .
Therefore the straignt lines AB, AC, are equal.
Each step of geometrical, and all other demonstra
tive reasoning, may be resolved into three such clauses
as these ; and these three clauses are termed respectively,
GROUNDS OF NECESSARY TRUTHS. 09
the major premiss, the minor premiss, and the conclu
sion; or, more briefly, the major, the minor, and the
conclusion.
The principle which justifies the reasoning when
exhibited in this syllogistic form, is this : — that a truth
which can be asserted as generally, or rather as univer
sally true, can be asserted as true also in each particular
case. The minor only asserts a certain particular case
to be an example of such conditions as are spoken of in
the major; and hence the conclusion, which is true of
the major by supposition, is true of the minor by conse
quence ; and thus we proceed from syllogism to syl
logism, in each one employing some general truth in
some particular instance. Any proof which occurs in
geometry, or any other science of demonstration, may
thus be reduced to a series of processes, in each of
which we pass from some general proposition to the
narrower and more special propositions which it in
cludes. And this process of deriving truths by the mere
combination of general principles, applied in particular
hypothetical cases, is called deduction; being opposed
to induction, in which, as we have seen, (Chap. I. Sect. 3.)
a new general principle is introduced at every step.
5. Now we have to remark that, this being so, how
ever far we follow such deductive reasoning, we can
never have, in our conclusion any truth which is not
virtually included in the original principles from which
the reasoning started. For since at any step we merely
take out of a general proposition something included in
it, while at the preceding step we have taken this ge
neral proposition out of one more general, and so on
perpetually, it is manifest that our last result was really
included in the principle or principles with which we
began. I say principles, because, although our logical
conclusion can only exhibit the legitimate issue of our
70 OF IDEAS IN GENEIIAL.
first principles, it may, nevertheless, contain the result
of the combination of several such principles, and may
thus assume a great degree of complexity, and may ap
pear so far removed from the parent truths, as to betray
at first sight hardly any relationship with them. Thus
the proposition which has already been quoted respect
ing the squares oii the sides of a right-angled triangle,
contains the results of many elementary principles ; as,
the definitions of parallels, triangle, and square ; the
axioms respecting straight lines, and respecting paral
lels; and, perhaps, others. The conclusion is compli
cated by containing the effects of the combination of all
these elements ; but it contains nothing, and can contain
nothing, but such elements and their combinations.
This doctrine, that logical reasoning produces no new
truths, but only unfolds and brings into view those truths
which were, in effect, contained in the first principles of
the reasoning, is assented to by almost all who, in
modern times, have attended to the science of logic.
Such a view is admitted both by those who defend, and
by those who depreciate the value of logic. " Whatever
is established by reasoning, must have been contained
and virtually asserted in the premises"'." "The only
truth which such propositions can possess consists in
conformity to the original principles."
In this manner the whole substance of our geometry
is reduced to the Definitions and Axioms which we
employ in our elementary reasonings ; and in like man
ner we reduce the demonstrative truths of any other
science to the definitions and axioms which we there
employ.
6. But in reference to this subject, it has sometimes
been said that demonstrative sciences do in reality depend
upon Definitions only; and that no additional kind of
* Whatelcy's Logic, pp. 237, 238.
GROUNDS OF NECESSARY TRUTHS. 71
principle, such as we have supposed Axioms to be, is
absolutely required. It has been asserted that in geo
metry, for example, the source of the necessary truth of
our propositions is this, that they depend upon definitions
alone, and consequently merely state the identity of the
same thing under different aspects.
That in the sciences which admit of demonstration,
as geometry, mechanics, and the like, Axioms as well as
Definitions are needed, in order to express the grounds
of our necessary convictions, must be shown hereafter
by an examination of each of these sciences in particular.
But that the propositions of these sciences, those of geo
metry for example, do not merely assert the identity of
the same thing, will, I think, be generally allowed, if we
consider the assertions which we are enabled to make.
When we declare that " a straight line is the shortest
distance between two points," is this merely an identical
proposition? the definition of a straight line in another
form ? Not so : the definition of a straight line involves
the notion of form only, and does not contain anything
about magnitude ; consequently, it cannot contain any
thing equivalent to " shortest." Thus the propositions
of geometry are not merely identical propositions ; nor
have we in their general character anything to coun
tenance the assertion, that they are the results of defi
nitions alone. And when we come to examine this and
other sciences more closely, we shall find that axioms,
such as are usually in our treatises made the funda
mental principles of our demonstrations, neither have
ever been, nor can be, dispensed with. Axioms, as well
as Definitions, are in all cases requisite, in order pro
perly to exhibit the grounds of necessary truth.
7. Thus the real logical basis of every body of demon
strated truths are the Definitions and Axioms which are
the first principles of the reasonings. But when we arc
72 OF IDEAS IX GENERAL.
arrived at this point, the question further occurs, what
is the ground of the truth of these Axioms? It is not
the logical, but the philosophical, not the formal, but the
real foundation of necessary truth, which we are seeking.
Hence this inquiry necessarily comes before us, What
is the ground of the Axioms of Geometry, of Mechanics,
and of any other demonstrable science ?
The answer which we are led to give, by the view
which we have taken of the nature of knowledge, has
already been stated. The ground of the axioms belong
ing to each science is the Idea which the axiom involves.
The ground of the Axioms of Geometry is the Idea of
Space: the ground of the Axioms of Mechanics is the
Idea of Force, of Action and Reaction, and the like. And
hence these Ideas are Fundamental Ideas ; and since they
are thus the foundations, not only of demonstration but
of truth, an examination into their real import and
nature is of the greatest consequence to our purpose.
8. Not only the Axioms, but the Definitions which
form the basis of our reasonings, depend upon our Fun
damental Ideas. And the Definitions are not arbitrary
definitions, but are determined by a necessity no less
rigorous than the Axioms themselves. We could not
think of geometrical truths without conceiving a circle ;
and we could not reason concerning such truths without
defining a circle in some mode equivalent to that which
is commonly adopted. The Definitions of parallels, of
right angles, and the like, are quite as necessarily pre
scribed by the nature of the case, as the Axioms which
these Definitions bring with them. Indeed we may
substitute one of these kinds of principles for another.
We cannot always put a Definition in the place of an
Axiom ; but we may always find an Axiom which shall
take the place of a Definition. If we assume a proper
Axiom respecting straight lines, we need no Definition
GROUNDS OF NECESSARY TRUTHS. 73
of a straight line. But in whatever shape the principle
appear, as Definition or as Axiom, it has about it nothing
casual or arbitrary, but is determined to be what it is, as
to its import, by the most rigorous necessity, growing
out of the Idea of Space.
9. These principles, — Definitions, and Axioms, — thus
exhibiting the primary developements of a fundamental
idea, do in fact express the idea, so far as its expression
in words forms part of our science. They are different
views of the same body of truth ; and though each prin
ciple, by itself, exhibits only one aspect of this body,
taken together they convey a sufficient conception of it
for our purposes. The Idea itself cannot be fixed in
words ; but these various lines of truth proceeding from
it, suggest sufficiently to a fitly-prepared mind, the place
where the idea resides, its nature, and its efficacy.
It is true that these principles, — our elementary Defi
nitions and Axioms, — even taken altogether, express the
Idea incompletely. Thus the Definitions and Axioms of
Geometry, as they are stated in our elementary works,
do not fully express the Idea of Space as it exists in our
minds. For, in addition to these, other Axioms, inde
pendent of these, and no less evident, can be stated ; and
are in fact stated when we come to the Higher Geo
metry. Such, for instance, is the Axiom of Archimedes
—that a curve line which joins two points is less than a
broken line which joins the same points and includes the
curve. And thus the Idea is disclosed but not fully re
vealed, imparted but not transfused, by the use we make
of it in science. When we have taken from the fountain
so much as serves our purpose, there still remains behind
a deep well of truth, which we have not exhausted, and
which we may easily believe to be inexhaustible.
CHAPTER VII.
THE FUNDAMENTAL IDEAS ARE NOT DERIVED
FROM EXPERIENCE.
1. BY the course of speculation contained in the last
three Chapters, we are again led to the conclusion which
we have already stated, that our knowledge contains an
ideal element, and that this element is not derived from
experience. For we have seen that there are proposi
tions which are known to be necessarily true ; and that
such knowledge is not, and cannot be, obtained by mere
observation of actual facts. It has been shown, also,
that these necessary truths are the results of certain fun
damental ideas, such as those of space, number, and the
like. Hence it follows inevitably that these ideas and
others of the same kind are not derived from experience.
For these ideas possess a power of infusing into their
developements that very necessity which experience can
in no way bestow. This power they do not borrow from
the external world, but possess by their own nature.
Thus we unfold out of the Idea of Space the propositions
of geometry, which are plainly truths of the most rigor
ous necessity and universality. But if the idea of space
were merely collected from observation of the external
world, it could never enable or entitle us to assert such
propositions : it could never authorize us to say that not
merely some lines, but all lines, not only have, but must
have, those properties which geometry teaches. Geo
metry in every proposition speaks a language which
experience never dares to utter; and indeed of which
she but half comprehends the meaning. Experience
sees that the assertions are true, but she sees' not how
profound and absolute is their truth. She unhesitatingly
assents to the laws which geometry delivers, but she does
FUNDAMENTAL IDEAS NOT DERIVATIVE. 75
not pretend to see the origin of their obligation. She
is always ready to acknowledge the sway of pure scien
tific principles as a matter of fact, but she does not
dream of offering her opinion on their authority as a
matter of right; still less can she justly claim to be her
self the source of that authority.
David Hume asserted'-, that we are incapable of
seeing in any of the appearances which the world pre
sents anything of necessary connexion ; and hence he
inferred that our knowledge cannot extend to any such
connexion. It will be seen from what we have said that
we assent to his remark as to the fact, but we differ from
him altogether in the consequence to be drawn from it.
Our inference from Hume's observation is, not the truth
of his conclusion, but the falsehood of his premises ;—
not that, therefore, we can know nothing of natural con
nexion, but that, therefore, we have some other source of
knowledge than experience : — not, that we can have no
idea of connexion or causation, because, in his language,
it cannot be the copy of an impression ; but that since
we have such an idea, our ideas are not the copies of
our impressions.
Since it thus appears that our fundamental ideas are
not acquired from the external world by our senses, but
have some separate and independent origin, it is im
portant for us to examine their nature and properties, as
they exist in themselves; and this it will be our business
to do through a portion of the following pages. But it
may be proper first to notice one or two objections
which may possibly occur to some readers.
2. It may be said that without the use of our senses,
of sight and touch, for instance, we should never have
any idea of space ; that this idea, therefore, may properly
be said to be derived from those senses. And to this I
* Essays, Vol. u. p. 70.
76 OF IDEAS IN GENERAL.
reply, by referring to a parallel instance. Without light
we should have no perception of visible figure ; yet the
power of perceiving visible figure cannot be said to be
derived from the light, but resides in the structure of the
eye. If we had never seen objects in the light, we
should be quite unaware that we possessed a power of
vision ; yet we should not possess it the less on that
account. If we had never exercised the senses of sight
and touch (if we can conceive such a state of human ex
istence) we know not that we should be conscious of an
idea of space. But the light reveals to us at the same
time the existence of external objects and our own power
of seeing. And in a very similar -manner, the exercise
of our senses discloses to us, at the same time, the ex
ternal world, and our own ideas of space, time, and other
conditions, without which the external world can neither
be observed nor conceived. That light is necessary to
vision, does not, in any degree, supersede the importance
of a separate examination of the laws of our visual
powers, if we would understand the nature of our own
bodily faculties and the extent of the information they
can give us. In like manner, the fact that intercourse
with the external world is necessary for the conscious
employment of our ideas, does not make it the less es
sential for us to examine those ideas in their most inti
mate structure, in order that \ve may understand the
grounds and limits of our knowledge. Even before we
see a single object, we have a faculty of vision ; and in
like manner, if we can suppose a man who has never
contemplated an object in space or time, we must still
assume him to have the faculties of entertaining the ideas
of space and time, which faculties are called into play
on the very first occasion of the use of the senses.
3. In answer to such remarks as the above, it has
sometimes been said that to assume separate faculties in
FUNDAMENTAL IDEAS NOT DERIVATIVE. 77
the mind for so many different processes of thought, is to
give a mere verbal explanation, sinee we learn nothing
concerning our idea of space by being told that we have
a faculty of forming such an idea. It has been said that
this course of explanation leads to an endless multipli
cation of elements in man's nature, without any advan
tage to our knowledge of his true constitution. We
may, it is said, assert man to have a faculty of walking,
of standing, of breathing, of speaking ; but what, it is
asked, is gained by such assertions? To this I reply, that
we undoubtedly have such faculties as those just named;
that it is by no means unimportant to consider them; and
that the main question in such cases is, whether they are
separate and independent faculties, or complex and deri
vative ones ; and, if the latter be the case, what are the
simple and original faculties by the combination of which
the others are produced. In walking, standing, breath
ing, for instance, a great part of the operation can be
reduced to one single faculty ; the voluntary exercise of
our muscles. But in breathing this does not appear to
be the whole of the process. The operation is, in part at
least, involuntary ; and it has been held that there is a
certain sympathetic action of the nerves, in addition to
the voluntary agency which they transmit, which is essen
tial to the function. To determine whether or no this
sympathetic faculty is real and distinct, and if so, what
are its laws and limits, is certainly a highly philosophical
inquiry, and well deserving the attention which has been
bestowed upon it by eminent physiologists. And just of
the same nature are the inquiries with respect to man's
intellectual constitution, on which we propose to enter.
For instance, man has a faculty of apprehending time,
and a faculty of reckoning numbers: arc these distinct, or
is one faculty derived from the other? To analy/e the
various combinations of our ideas and observations into
78 OF IDEAS IN GENERAL.
the original faculties which they involve ; to show that
these faculties are original, and not capable of further
analysis : to point out the characters which mark these
faculties and lead to the most important features of our
knowledge; — these are the kind of researches on which
we have now to enter, and these, we trust, will be found
to be far from idle or useless parts of our plan. If we
succeed in such attempts, it will appear that it is by
no means a frivolous or superfluous step to distinguish
separate faculties in the mind. If we do not learn much
by being told that we have a faculty of forming the idea
of space, wre at least, by such a commencement, circum
scribe a certain portion of the field of our investigations,
which, we shall afterwards endeavour to show, requires
and rewards a special examination. And though we shall
thus have to separate the domain of our philosophy into
many provinces, these are, as we trust it will appear,
neither arbitrarily assigned, nor vague in their limits,
nor infinite in number.
CHAPTER VIII.
OF THE PHILOSOPHY OF THE SCIENCES.
WE proceed, in the ensuing Books, to the closer exami
nation of a considerable number of those Fundamental
Ideas on which the sciences, hitherto most successfully
cultivated, are founded. In this task, our objects will
be to explain and analyze such Ideas so as to bring into
view the Definitions and Axioms, or other forms, in
which we may clothe the conditions to which our specu
lative knowledge is subjected. I shall also try to prove,
for some of these Ideas in particular, what has been
already urged respecting them in general, that they are
PHILOSOPHY OF SCIENCES. 79
not derived from observation, but necessarily impose
their conditions upon that knowledge of which observa
tion supplies the materials. I shall further, in some
cases, endeavour to trace the history of these Ideas as
they have successively come into notice in the progress
of science; the gradual developement by which they have
arrived at their due purity and clearness; and, as a
necessary part of such a history, I shall give a view of
some of the principal controversies which have taken
place with regard to each portion of knowledge.
An exposition and discussion of the Fundamental
Ideas of each Science may, with great propriety, be
termed the PHILOSOPHY OF such SCIENCE. These ideas
contain in themselves the elements of those truths which
the science discovers and enunciates; and in the progress
of the sciences, both in the world at large and in the
mind of each individual student, the most important
steps consist in apprehending these ideas clearly, and in
bringing them into accordance with the observed facts.
I shall, therefore, in a series of Books, treat of the Phi
losophy of the Pure Sciences, the Philosophy of the
Mechanical Sciences, the Philosophy of Chemistry, and
the like, and shall analyze and examine the ideas which
these sciences respectively involve.
In this undertaking, inevitably somewhat long, and
involving many deep and subtle discussions, I shall take,
as a chart of the country before me, by which my course
is to be guided, the scheme of the sciences which I was
led to form by travelling over the history of each in
order*. Each of the sciences of which I then narrated
the progress, depends upon several of the Fundamental
Ideas of which I have to speak : some of these Ideas are
peculiar to one field of speculation, others are common
to more. A previous enumeration of Ideas thus collected
* History of lite Inductive Sciences.
80 OF IDEAS IN GENERAL.
may serve both to show the course and limits of this part
of our plan, and the variety of interest which it offers.
I shall, then, successively, have to speak of the Ideas
which are the foundation of Geometry and Arithmetic,
(and which also regulate all sciences depending upon
these, as Astronomy and Mechanics;) namely, the Ideas
of Space, Time, and Number :
Of the Ideas on which the Mechanical Sciences (as
Mechanics, Hydrostatics, Physical Astronomy) more pecu
liarly rest ; the ideas of Force and Matter •, or rather the
idea of Cause, which is the basis of these :
Of the Ideas which the Secondary Mechanical Sciences
(Acoustics, Optics, and Thermotics) involve ; namely, the
Ideas of the Externality of objects, and of the Media
by which we perceive their qualities :
Of the Ideas which are the basis of Mechanico-chc-
mical and Chemical Science; Polarity, Chemical Affinity,
and Substance ; and the Idea of Symmetry, a necessary
part of the Philosophy of Crystallography :
Of the Ideas on which the Classificatory Sciences
proceed (Mineralogy, Botany, and Zoology) ; namely, the
Ideas of Resemblance, and of its gradations, and of
Natural Affinity:
Finally, of those Ideas on which the Physiological
Sciences are founded ; the Ideas of separate Vital Powers,
such as Assimilation and Irritability ; and the Idea of
Final Cause.
We have, besides these, the Palsetiological Sciences,
which proceed mainly on the conception of Historical
Causation.
It is plain that when we have proceeded so far as
this, we have advanced to the verge of those speculations
which have to do with mind as well as body. The
extension of our philosophy to such a field, if it can be
justly so extended, will be one of the most important
PHILOSOPHY OF SCIENCES. 81
results of our researches; but on that very account we
must fully study the lessons which we learn in those
fields of speculation where our doctrines are most secure,
before we venture into a region where our principles will
appear to be more precarious, and where they are inevi
tably less precise.
We now proceed to the examination of the above
Ideas, and to such essays towards the philosophy of each
Science as this course of investigation may suggest.
VOL. i. w. P.
G
82
BOOK II.
THE PHILOSOPHY OF THE PURE
SCIENCES,
CHAPTER I.
OF THE PURE SCIENCES.
1. ALL external objects and events which we can con
template are viewed as having relations of Space, Time,
and Number ; and are subject to the general conditions
which these Ideas impose, as well as to the particular
laws which belong to each class of objects and occur
rences. The special laws of nature, considered under
the various aspects which constitute the different sciences,
are obtained by a mixed reference to experience and to
the fundamental ideas of each science. But besides the
sciences thus formed by the aid of special experience, the
conditions which flow from those more comprehensive
ideas first mentioned, Space, Time, and Number, consti
tute a body of science, applicable to objects and changes
of all kinds, and deduced without recurrence being had
to any observation in particular. These sciences, thus
unfolded out of ideas alone, unmixed with any reference
to the phenomena of matter, are hence termed Pure
Sciences. The principal sciences of this class are Geome
try, Theoretical Arithmetic, and Algebra considered in its
most general sense, as the investigation of the relations
of space and number by means of general symbols.
OF THE PURE SCIENCES. 83
2. These Pure Sciences were not included in our
survey of the history of the sciences, because they are
not inductive sciences. Their progress has riot consisted
in collecting laws from phenomena, true theories from
observed facts, and more general from more limited laws ;
but in tracing the consequences of the ideas themselves,
and in detecting the most general and intimate analogies
and connexions which prevail among such conceptions as
are derivable from the ideas. These sciences have no
principles besides definitions and axioms, and no process
I of proof but deduction ; this process, however, assuming
here a most remarkable character ; and exhibiting a com
bination of simplicity and complexity, of rigour and
generality, quite unparalleled in other subjects.
3. The universality of the truths, and the rigour of
the demonstrations of these pure sciences, attracted
attention in the earliest times ; and it was perceived that
they offered an exercise and a discipline of the intellec
tual faculties, in a form peculiarly free from admixture
of extraneous elements. They were strenuously culti
vated by the Greeks, both with a view to such a disci
pline, and from the love of speculative truth which pre
vailed among that people : and the name mathematics, by
which they are designated, indicates this their character
of disciplinal studies.
4. As has already been said, the ideas which these
sciences involve extend to all the objects and changes
which we observe in the external world ; and hence the
consideration of mathematical relations forms a large
portion of many of the sciences which treat of the phe
nomena and laws of external nature, as Astronomy,
Optics, and Mechanics. Such sciences are hence often
termed Mixed Mathematics, the relations of space and
number being, in these branches of knowledge, combined
with principles collected from special observation ;
84 PHILOSOPHY OF THE PURE SCIENCES.
while Geometry, Algebra, and the like subjects, which
involve no result of experience, are called Pure Mathe
matics.
5. Space, time, and number, may be conceived as
forms by which the knowledge derived from our sensa
tions is moulded, and which are independent of the dif
ferences in the matter of our knowledge, arising from the
sensations themselves. Hence the sciences which have
these ideas for their subject may be termed Formal
Sciences. In this point of view, they are distinguished
from sciences in which, besides these mere formal laws
by which appearances are corrected, we endeavour to
apply to the phenomena the idea of cause, or some of the
other ideas which penetrate further into the principles
of nature. We have thus, in the History, distinguished
Formal Astronomy and Formal Optics from Physical
Astronomy and Physical Optics.
We now proceed to our examination of the Ideas
which constitute the foundation of these formal or pure
mathematical sciences, beginning with the Idea of Space.
CHAPTER II.
OF THE IDEA OF SPACE.
1. BY speaking of space as an Idea, I intend to imply,
as has already been stated, that the apprehension of
objects as existing in space, and of the relations of posi
tion, &c., prevailing among them, is not a consequence
of experience, but a result of a peculiar constitution and
activity of the mind, which is independent of all expe
rience in its origin, though constantly combined with
experience in its exercise.
That the idea of space is thus independent of experi
ence, has already been pointed out in speaking of ideas
OF THE IDEA OF SPACE. 85
in general : but it may be useful to illustrate the doctrine
further in this particular case.
I assert, then, that space is not a notion obtained
by experience. Experience gives us information con
cerning things without us : but our apprehending them
as without us, takes for granted their existence in space.
Experience acquaints us what are the form, position,
magnitude of particular objects : but that they have form,
position, magnitude, presupposes that they are in space.
We cannot derive from appearances, by the way of
observation, the habit of representing things to ourselves
as in space ; for no single act of observation is possible
any otherwise than by beginning with such a representa
tion, and conceiving objects as already existing in space.
2. That our mode of representing space to ourselves
is not derived from experience, is clear also from this :
— that through this mode of representation we arrive at
propositions which are rigorously universal and neces
sary. Propositions of such a kind could not possibly be
obtained from experience ; for experience can only teach
us by a limited number of examples, and therefore can
never securely establish a universal proposition : and
again, experience can only inform us that anything is so,
and can never prove that it must be so. That two sides
of a triangle are greater than the third is a universal
and necessary geometrical truth: it is true of all tri
angles ; it is true in such a way that the contrary cannot
be conceived. Experience could not prove such a propo
sition. And experience has not proved it ; for perhaps
no man ever made the trial as a means of removing
doubts : and no trial could, in fact, add in the smallest
degree to the certainty of this truth. To seek for proof
of geometrical propositions by an appeal to observation
proves nothing in reality, except that the person who
has recourse to such grounds has no due apprehension
86 PHILOSOPHY OF THE PURE SCIENCES.
of the nature of geometrical demonstration. We have
heard of persons who convinced themselves by measure
ment that the geometrical rule respecting the squares
on the sides of a right-angled triangle was true : but
these were persons whose minds had been engrossed by
practical habits, and in whom the speculative develope-
ment of the idea of space had been stifled by other em
ployments. The practical trial of the rule may illustrate,
but cannot prove it. The rule will of course be con
firmed by such trial, because what is true in general is
true in particular: but the rule cannot be proved from any
number of trials, for no accumulation of particular cases
makes up a universal case. To all persons who can see
the force of any proof, the geometrical rule above referred
to is as evident, and its evidence as independent of ex
perience, as the assertion that sixteen and nine make
twenty-five. At the same time, the truth of the geome
trical rule is quite independent of numerical truths, and
results from the relations of space alone. This could
not be if our apprehension of the relations of space were
the fruit of experience : for experience has no element
from which such truth and such proof could arise.
3. Thus the existence of necessary truths, such as
those of geometry, proves that the idea of space from
which they flow, is not derived from experience. Such
truths are inconceivable on the supposition of their being
collected from observation ; for the impressions of sense
include no evidence of necessity. But we can readily
understand the necessary character of such truths, if we
conceive that there are certain necessary conditions under
which alone the mind receives the impressions of sense.
Since these conditions reside in the constitution of the
mind, and apply to every perception of an object to
which the mind can attain, we easily see that their rules
must include, not only all that has been, but all that can
OF THE IDEA OF SPACE. 87
be, matter of experience. Our sensations can each con
vey no information except about itself; each can contain
no trace of another additional sensation ; and thus no
relation and connexion between two sensations can be
given by the sensations themselves. But the mode in
which the mind perceives these impressions as objects,
may and will introduce necessary relations among them :
and thus by conceiving the idea of space to be a con
dition of perception in the mind, we can conceive the
existence of necessary truths, which apply to all per
ceived objects.
4. If we consider the impressions of sense as the
mere materials of our experience, such materials may
be accumulated in any quantity and in any order. But
if we suppose that this matter has a certain form given
it, in the act of being accepted by the mind, we can
understand how it is that these materials are subject to
inevitable rules ; — how nothing can be perceived exempt
from the relations which belong to such a form. And
since there are such truths applicable to our experience,
and arising from the nature of space, we may thus
consider space as a form which the materials given by
experience necessarily assume in the mind; as an ar
rangement derived from the perceiving mind, and not
from the sensations alone.
5. Thus this phrase, — that space is &form belonging
to our perceptive power, — may be employed to express
that we cannot perceive objects as in space, without an
operation of the mind as well as of the senses — without
active as well as passive faculties. This phrase, how
ever, is not necessary to the exposition of our doctrines.
Whether we call the conception of space a condition of
perception, a form of perception, or an idea, or by any
other term, it is something originally inherent in the
mind perceiving, and not in the objects perceived. And
88 PHILOSOPHY OF THE PURE SCIENCES.
it is because the apprehension of all objects is thus sub
jected to certain mental conditions, forms or ideas, that
our knowledge involves certain inviolable relations and
necessary truths. The principles of such truths, so far
as they regard space, are derived from the idea of space>
and we must endeavour to exhibit such principles in
their general form. But before we do this, we may
notice some of the conditions which belong, not to our
Ideas in general, but to this Idea of Space in parti
cular.
CHAPTER III.
OF SOME PECULARITIES OF THE IDEA OF
SPACE.
1. SOME of the Ideas which we shall have to examine
involve conceptions of certain relations of objects, as the
idea of Cause and of Likeness ; and may appear to be
suggested by experience, enabling us to abstract this
general relation from particular cases. But it will be
seen that Space is not such a general conception of a
relation. For we do not speak of Spaces as we speak of
Causes and Likenesses, but of Space. And when wre
speak of spaces, we understand by the expression, parts
of one and the same identical every where -extended
Space. We conceive a Universal Space ; which is not
made up of these partial spaces as its component parts,
for it would remain if these were taken away ; and these
cannot be conceived without presupposing absolute space.
Absolute Space is essentially one ; and the complication
which exists in it, and the conception of various spaces,
depends merely upon boundaries. Space must, there
fore, be, as we have said, not a general conception
abstracted from particulars, but a universal mode of
representation, altogether independent of experience.
PECULIARITIES OF THE IDEA OF SPACE. 89
2. Space is infinite. We represent it to ourselves as
an infinitely great magnitude. Such an idea as that of
Likeness or Cause, is, no doubt, found in an infinite
number of particular cases, and so far includes these
cases. But these ideas do riot include an infinite number
of cases as parts of an infinite whole. When we say
that all bodies and partial spaces exist in infinite space,
we use an expression which is not applied in the same
sense to any cases except those of Space and Time.
3. What is here said may appear to be a denial of
the real existence of space. It must be observed, how
ever, that we do not deny, but distinctly assert, the
existence of space as a real and necessary condition of
all objects perceived ; and that we not only allow that
objects are seen external to us, but we found upon the
fact of their being so seen, our view of the nature of
space. If, however, it be said that we deny the reality
of space as an object or thing, this is true. Nor does it
appear easy to maintain that space exists as a thing,
when it is considered that this thing is infinite in all its
dimensions; and, moreover, that it is a thing, which,
being nothing in itself, exists only that other things may
exist in it. And those who maintain the real existence
of space, must also maintain the real existence of time in
the same sense. Now two infinite things, thus really
existing, and yet existing only as other things exist in
them, are notions so extravagant that we are driven to
some other mode of explaining the state of the matter.
4. Thus space is not an object of which we perceive
the properties, but a form of our perception; not a thing
which aifects our senses, but an idea to which we con
form the impressions of sense. And its peculiarities ap
pear to depend upon this, that it is not only a form of
sensation, but of intuition ; that in reference to space,
we not only perceive but contemplate objects. We see
90 PHILOSOPHY OF THE PURE SCIENCES.
objects in space, side by side, exterior to each other;
space, and objects in so far as they occupy space, have
parts exterior to other parts ; and have the whole thus
made up by the juxtaposition of parts. This mode of
apprehension belongs only to the ideas of space and
time. Space and Time are made up of parts, but Cause
and Likeness are not apprehended as made up of parts.
And the term intuition (in its rigorous sense) is appli
cable only to that mode of contemplation in which we
thus look at objects as made up of parts, and apprehend
the relations of those parts at the same time and by the
same act by which we apprehend the objects themselves.
5. As we have said, space limited by boundaries gives
rise to various conceptions which we have often to con
sider. Thus limited, space assumes/brw or figure; and
the variety of conceptions thus brought under our notice
is infinite. We have every possible form of line, straight
line, and curve ; and of curves an endless number ; — cir
cles, parabolas, hyperbolas, spirals, helices. We have
plane surfaces of various shapes, — parallelograms, poly
gons, ellipses ; and we have solid figures, — cubes, cones,
cylinders, spheres, spheroids, and so on. All these have
their various properties, depending on the relations of
their boundaries ; and the investigation of their proper
ties forms the business of the science of Geometry.
6. Space has three dimensions, or directions in which
it may be measured ; it cannot have more or fewer. The
simplest measurement is that of a straight line, which
has length alone. A surface has both length and
breadth : and solid space has length, breadth, and thick
ness or depth. The origin of such a difference of dimen
sions will be seen if we reflect that each portion of space
lias a boundary, and is extended both in the direction in
which its boundary extends, and also in a direction from
its boundary ; for otherwise it would not be a boundary.
PECULARITIES OF THE IDEA OF SPACE. 91
A point has no dimensions. A line has but one dimen
sion, — the distance from its boundary, or its length. A
plane, bounded by a straight line, has the dimension
which belongs to this line, and also has another dimen
sion arising from the distance of its parts from this bound
ary line ; and this may be called breadth. A solid,
bounded by a plane, has the dimensions which this plane
has ; and has also a third dimension, which we may call
height or depth, as we consider the solid extended above
or below the plane ; or thickness, if we omit all con
sideration of up and down. And no space can have any
dimensions which are not resoluble into these three.
We may now proceed to consider the mode in which
the idea of space is employed in the formation of
Geometry.
CHAPTER IV.
OF THE DEFINITIONS AND AXIOMS WHICH
RELATE TO SPACE.
1. THE relations of space have been apprehended
with peculiar distinctness and clearness from the very
first unfolding of man's speculative powers. This was a
consequence of the circumstance which we have just
noticed, that the simplest of these relations, and those on
which the others depend, are seen by intuition. Hence,
as soon as men were led to speculate concerning the
relations of space, they assumed just principles, and
obtained true results. It is said that the science of
geometry had its origin in Egypt, before the dawn of the
Greek philosophy : but the knowledge of the early
Egyptians (exclusive of their mythology) appears to have
been purely practical ; and, probably, their geometry
consisted only in some maxims of la nd-measuring, which
is what the term implies. The Greeks of the time of
92 PHILOSOPHY OF THE PURE SCIENCES.
Plato, had, however, not only possessed themselves of
many of the most remarkable elementary theorems of
the science ; but had, in several instances, reached the
boundary of the science in its elementary form ; as when
they proposed to themselves the problems of doubling
the cube and squaring the circle.
But the deduction of these theorems by a systematic
process, and the primary exhibition of the simplest prin
ciples involved in the idea of space, which such a
deduction requires, did not take place, so far as we are
aware, till a period somewhat later. The Elements of
Geometry of Euclid, in which this task was performed,
are to this day the standard work on the subject : the
author of this work taught mathematics with great
applause at Alexandria, in the reign of Ptolemy Lagus,
about 280 years before Christ. The principles which
Euclid makes the basis of his system have been very
little simplified since his time ; and all the essays and
controversies which bear upon these principles, have
had a reference to the form in which they are stated
by him.
2. Definitions. — The first principles of Euclid's geo
metry are, as the first principles of any system of
geometry must be, definitions and axioms respecting
the various ideal conceptions which he introduces; as
straight lines, parallel lines, angles, circles, and the like.
But it is to be observed that these definitions and
axioms are very far from being arbitrary hypotheses and
assumptions. They have their origin in the idea of
space, and are merely modes of exhibiting that idea in
such a manner as to make it afford grounds of deductive
reasoning. The axioms are necessary consequences of
the conceptions respecting which they are asserted ; and
the definitions are no less necessary limitations of con
ceptions ; not requisite in order to arrive at this or that
DEFINITIONS AND AXIOMS RELATING TO SPACE. 93
consequence ; but necessary in order that it may be
possible to draw any consequences, and to establish any
general truths.
For example, if we rest the end of one straight
staff upon the middle of another straight staff, and move
the first staff into various positions, we, by so doing,
alter the angles which the first staff makes with the
other to the right hand and to the left. But if we
place the staff in that special position in which these
two angles are equal, each of them is a right angle,
according to Euclid ; and this is the definition of a right
angle, except that Euclid employs the abstract con
ception of straight lines, instead of speaking, as we have
done, of staves. But this selection of the case in which
the two angles are equal is not a mere act of caprice ;
as it might have been if he had selected a case in which
these angles are unequal in any proportion. For the
consequences which can be drawn concerning the cases
of unequal angles, do not lead to general truths, without
some reference to that peculiar case in which the angles
are equal : and thus it becomes necessary to single out
and define that special case, marking it by a special
phrase. And this definition not only gives complete and
distinct knowledge what a right angle is, to any one
who can form the conception of an angle in general ; but
also supplies a principle from which all the properties of
right angles may, be deduced.
3. Axioms.— With regard to other conceptions also,
as circles, squares, and the like, it is possible to lay
down definitions which are a sufficient basis for our
reasoning, so far as such figures are concerned. But,
besides these definitions, it has been found necessary to
introduce certain axioms among the fundamental prin
ciples of geometry. These are of the simplest character ;
for instance, that two straight lines cannot cut each
94 PHILOSOPHY OF THE PUKE SCIENCES.
other in more than one point, and an axiom concerning
parallel lines. Like the definitions, these axioms flow
from the Idea of Space, and present that idea under
various aspects. They are different from the definitions ;
nor can the definitions be made to take the place of the
axioms .in the reasoning by which elementary geo
metrical properties are established. For example, the
definition of parallel straight lines is, that they are such
as, however far continued, can never meet : but, in order
to reason concerning such lines, we must further adopt
some axiom respecting them : for example, we may very
conveniently take this axiom ; that two straight lines
which cut one another are not both of them parallel to
a third straight line"". The definition and the axiom are
seen to be inseparably connected by our intuition of the
properties of space ; but the axiom cannot be proved
from the definition, by any rigorous deductive demon
stration. And if we were to take any other definition of
two parallel straight lines, (as that they are both per
pendicular to a third straight line,) we should still, at
some point or other of our progress, fall in with the
same difficulty of demonstratively establishing their pro
perties without some further assumption.
4. Thus the elementary properties of figures, which
are the basis of our geometry, are necessary results of
our Idea of Space ; and are connected with each other
by the nature of that idea, and not merely by our hypo
theses and constructions. Definitions and axioms must
be combined, in order to express this idea so far as
the purposes of demonstrative reasoning require. These
verbal enunciations of the results of the idea cannot be
made to depend on each other by logical consequence ;
but have a mutual dependence of a more intimate kind,
* This axiom is simpler and more convenient than that of Euclid.
It is employed by the late Professor Flayfair in his Geometry.
DEFINITIONS AND AXIOMS RELATING TO SPACE. 95
which words cannot fully convey. It is not possible to
resolve these truths into certain hypotheses, of which all
the rest shall be the necessary logical consequence. The
necessity is not hypothetical, but intuitive. The axioms
require not to be granted, but to be seen. If any one
were to assent to them without seeing them to be true,
his assent would be of no avail for purposes of reason
ing: for he would be also unable to see in what cases
they might be applied. The clear possession of the
Idea of Space is the first requisite for all geometrical
reasoning ; and this clearness of idea may be tested by
examining whether the axioms offer themselves to the
mind as evident.
5. The necessity of ideas added to sensations, in
order to produce knowledge, has often been overlooked
or denied in modern times. The ground of necessary
truth which ideas supply being thus lost, it was con
ceived that there still remained a ground of necessity in.
definitions; — that we might have necessary truths, by
asserting especially what the definition implicity involved
in general. It was held, also, that this was the case in
geometry: — that all the properties of a circle, for
instance, were implicitly contained in the definition of a
circle. That this alone is not the ground of the neces
sity of the truths which regard the circle, — that we
could not in this way unfold a definition into propor
tions, without possessing an intuition of the relations to
which the definition led, — has already been shown. But
the insufficiency of the above account of the grounds of
necessary geometrical truth appeared in another way
also. It was found impossible to lay down a system of
definitions out of which alone the whole of geometrical
truth could be evolved. It was found that axioms could
not be superseded. No definition of a straight line
could be given which rendered the axiom concerning
96 PHILOSOPHY OF THE PURE SCIENCES.
straight lines superfluous. And thus it appeared that
the source of geometrical truths was not definition
alone ; and we find in this result a confirmation of the
doctrine which we are here urging, that this source of
truth is to be found in the form or conditions of our
perception ; — in the idea which we unavoidably combine
with the impressions of sense ; — in the activity, and not
in the passivity of the mind"-.
6. This will appear further when we come to con
sider the mode in which \ve exercise our observation
upon the relations of space. But we may, in the first
place, make a remark which tends to show the con
nexion between our conception of a straight line, and
the axiom which is made the foundation of our reason
ings concerning space. The axiom is this; — that two
straight lines, which have both their ends joined, cannot
have the intervening parts separated so as to inclose a
space. The necessity of this axiom is of exactly the
same kind as the necessity of the definition of a right
angle, of which we have already spoken. For as the line
standing on another makes right angles when it makes
the angles on the two sides of it equal; so a line is a
straight line when it makes the two portions of space,
on the two sides of it, similar. And as there is only a
single position of the line first mentioned, which can
make the angles equal, so there is only a single form of
a line which can make the spaces near the line similar
on one side and on the other : and therefore there can
not be two straight lines, such as the axiom describes,
* I formerly stated views similar to these in some " Remarks"
appended to a work which I termed The Mechanical Euclid, pub
lished in 1837- These Remarks, so far as they bear upon the question
here discussed, were noticed and controverted in No. 135 of the Edin
burgh Review. As an examination of the reviewer's objections may
serve further to illustrate the subject, I shall annex to this chapter an
answer to the article to which I have referred.
DEFINITIONS AND AXIOMS RELATING TO SPACE. 07
which, between the same limits, give t\vo different
boundaries to space thus separated. And thus we see a
reason for the axiom. Perhaps this view may be further
elucidated if we take a leaf of paper, double it, and
crease the folded edge. We shall thus obtain a straight
line at the folded edge ; and this line divides the surface
of the paper, as it was originally spread out, into two
similar spaces. And that these spaces are similar so far
as the fold which separates them is concerned, appears
from this ; — that these two parts coincide when the
paper is doubled. And thus a fold in a sheet of paper
at the same time illustrates the definition of a straight
line according to the above view, and confirms the
axiom that two such lines cannot enclose a space.
If the separation of the two parts of space were made
by any other than a straight line; if, for instance, the
paper were cut by a concave line ; then, on turning one
of the parts over, it is easy to see that the edge of one
part being concave one way, and the edge of the other
part concave the other way, these two lines would
enclose a space. And each of them would divide the
whole space into two portions which were not similar ;
for one portion would have a concave edge, and the
other a convex edge. Between any two points, there
might be innumerable lines drawn, some, convex one
way, and some, convex the other way ; but the straight
line is the line which is not convex either one way or
the other ; it is the single medium standard from which
the others may deviate in opposite directions.
Such considerations as these show sufficiently that
the singleness of the straight line which connects any
two points is a result of our fundamental conceptions of
space. But yet the above conceptions of the similar
form of the two parts of space on the two sides of a line,
and of the form of a line which is intermediate among
VOL. i. w. p. H
98 PHILOSOPHY OF THE PURE SCIENCES.
all other forms, are of so vague a nature, that they can
not fitly be made the basis of our elementary geometry ;
and they are far more conveniently replaced, as they
have been in almost all treatises of geometry, by the
axiom, that two straight lines cannot inclose a space.
7. But we may remark that, in what precedes, we
have considered space only under one of its aspects : — as
a plane. The sheet of paper wrhich we assumed in order
to illustrate the nature of a straight line, was supposed
to be perfectly plane m flat : for otherwise, by folding it,
we might obtain a line not straight. TSow this assump
tion of a plane appears to take for granted that very
conception of a straight line which the sheet was em
ployed to illustrate ; for the definition of a plane given
in the Elements of Geometry is, that it is a surface on
which lie all straight lines drawn from one point of the
surface to another. And thus the explanation above
given of the nature of a straight line, — that it divides a
plane space into similar portions on each side, — appears
to be imperfect or nugatory.
To this we reply, that the explanation must be ren
dered complete and valid by deriving the conception of
a plane from considerations of the same kind as those
which we employed for a straight line. Any portion of
solid space may be divided into two portions by surfaces
passing through any given line or boundaries. And
these surfaces may be convex either on one side or on
the other, and they admit of innumerable changes from
being convex on one side to being convex on the other
in any degree. So long as the surface is convex either
way, the two portions of space which it separates are not
similar, one having a convex and the other a concave
boundary. But there is a certain intermediate position of
the surface, in which position the two portions of space
which it divides have their boundaries exactly similar.
DEFINITIONS AND AXIOMS RELATING TO SPACE. 90
Iii this position, the surface is neither convex nor concave,
but plane. And thus a plane surface is determined by
this condition — of its being that single surface which is
the intermediate form among all convex and concave
surfaces by which solid space can be divided, — and of
its separating such space into two portions, of which
the boundaries, though they are the same surface in
two opposite positions, are exactly similar.
Thus a plane is the simplest and most symmetrical
boundary by which a solid can be divided ; and a straight
line is the simplest and most symmetrical boundary by
which a plane can be separated. These conceptions are
obtained by considering the boundaries of an intermin
able space, capable of imaginary division in every direc
tion. And as a limited space may be separated into two
parts by a plane, and a plane again separated into two
parts by a straight line, so a line is divided into two por
tions by a point, which is the common boundary of the
t\vo portions ; the end of the one and the beginning of the
other portion having itself no magnitude, form, or parts.
8. The geometrical properties of planes and solids
are deducible from the first principles of the Elements,
without any new axioms ; the definition of a plane above
quoted, — that all straight lines joining its points lie in
the plane, — being a sufficient basis for all reasoning upon
these subjects. And thus, the views which we have pre
sented of the nature of space being verbally expressed
by means of certain definitions and axioms, become the
groundwork of a long series of deductive reasoning, by
which is established a very large and curious collection
of truths, namely, the whole science of Elementary
Plane and Solid Geometry.
This science is one of indispensable use and constant
reference, for every student of the laws of nature ; for the
relations of space and number are the alphabet in which
II 2
100 PHILOSOPHY OF THE PUKE SCIENCES.
those laws are written. But besides the interest and im
portance of this kind which geometry possesses, it has a
great and peculiar value for all who wish to understand
the foundations of human knowledge, and the methods
by which it is acquired. For the student of geometry
acquires, with a degree of insight and clearness which
the unmathematical reader can but feebly imagine, a
conviction that there are necessary truths, many of them
of a very complex and striking character ; and that a
few of the most simple and self-evident truths which it is
possible for the mind of man to apprehend, may, by
systematic deduction, lead to the most remote and unex
pected results.
In pursuing such philosophical researches as that
in which we are now engaged, it is of great advantage
to the speculator to have cultivated to some extent the
study of geometry ; since by this study he may become
fully aware of such features in human knowledge as
those which we have mentioned. By the aid of the
lesson thus learned from the contemplation of geome
trical truths, wre have been endeavouring to establish
those further doctrines; — that these truths are but dif
ferent aspects of the same Fundamental Idea, and that
the grounds of the necessity which these truths possess
reside in the Idea from which they flow, this Idea not
being a derivative result of experience, but its primary
rule. When the reader has obtained a clear and satis
factory view of these doctrines, so far as they are appli
cable to our knowledge concerning space, he has, we may
trust, overcome the main difficulty which will occur in
following the course of the speculations now presented
to him. He is then prepared to go forwards with us ; to
see over how wide a field the same doctrines are appli
cable: and how rich and various a harvest of knowledge
springs from these seemingly scanty principles.
DEFINITIONS AND AXIOMS RELATING TO SPACE. 101
But before we quit the subject now under our con
sideration, we shall endeavour to answer some objections
which have been made to the views here presented; and
shall attempt to illustrate further the active powers which
we have ascribed to the mind.
CHAPTER V.
OF SOME OBJECTIONS WHICH HAVE BEEN
MADE TO THE DOCTRINES STATED
IN THE PREVIOUS CHAPTER"-.
THE Edinburgh Review, No. cxxxv., contains a cri
tique on a work termed The Mechanical Euclid, in which
opinions were delivered to nearly the same effect as some
of those stated in the last chapter, and in Chapter xi.
of the First Book. Although I believe that there are no
arguments used by the reviewer to which the answers
will not suggest themselves in the mind of any one who
has read with attention what has been said in the pre
ceding chapters (except, perhaps, one or two remarks
which have reference to mechanical ideas), it may serve to
* In order to render the present chapter more intelligible, it may
be proper to state briefly the arguments which gave occasion to the
review. After noticing Stewart's assertions, that the certainty of mathe
matical reasoning arises from its depending upon definitions, and that
mathematical truth is hypothetical; I urged, — that no one has yet
been able to construct a system of mathematical truths by the aid of
definitions alone ; that a definition would not be admissible or appli
cable except it agreed with a distinct conception in the mind ; that the
definitions which we employ in mathematics are not arbitrary or hypo
thetical, but necessary definitions; that if Stewart had taken as his
examples of axioms the peculiar geometrical axioms, his assertions
would have been obviously erroneous ; and that the real foundation of
the truths of mathematics is the Idea of Space, which may be expressed
(for purposes of demonstration) partly by definitions and partly by
axiom-.
102 PHILOSOPHY OF THE PURE SCIENCES.
illustrate the subject if I reply to the objections directly,
taking them as the reviewer has stated them.
1. I had dissented from Stewart's assertion that
mathematical truth is hypothetical, or depends upon arbi
trary definitions ; since we understand by an hypothesis
a supposition, not only which we may make, but may
abstain from making, or may replace by a different sup
position ; whereas the definitions and hypotheses of geo
metry are necessarily such as they are, and cannot be
altered or excluded. The reviewer (p. 84), informs us
that he understands Stewart, when he speaks of hypo
theses and definitions being the foundation of geometry,
to speak of the hypothesis that real objects correspond
to our geometrical definitions. " If a crystal be an exact
hexahedron, the geometrical properties of the hexahe
dron may be predicated of that crystal." To this I reply,
— that such hypotheses as this are the grounds of our
applications of geometrical truths to real objects, but
can in no way be said to be the foundation of the truths
themselves; — that I do not think that the sense which the
reviewer gives was Stewart's meaning; — but that if it was,
this view of the use of mathematics does not at all affect
the question which both he and I proposed to discuss,
which was, the ground of mathematical certainty. I may
add, that whether a crystal be an exact hexahedron, is
a matter of observation and measurement, not of defini
tion. I think the reader can have no difficulty in seeing
how little my doctrine is affected by the connexion on
which the reviewer thus insists. I have asserted that the
proposition which affirms the square on the diagonal of
a rectangle to be equal to the squares on two sides, does-
not rest upon arbitrary hypotheses; the objector answers,
that the proposition that the square on the diagonal of
this page is equal to the squares on the sides, depends
upon the arbitrary hypothesis that the page is a rect-
ANSWER TO OBJECTIONS. 103
angle. Even if this fact were a matter of arbitrary
hypothesis, what could it have to do with the general
geometrical proposition? How could a single fact, ob
served or hypothetical, affect a universal and necessary
truth, which would be equally true if the fact were false?
If there be nothing arbitrary or hypothetical in geometry
till we come to such steps in its application, it is plain
that the truths themselves are not hypothetical; which is
the question for us to decide.
2. The reviewer then (p. 85), considers the doctrine
that axioms as well as definitions are the foundations of
geometry; and here he strangely narrows and confuses
the discussion by making himself the advocate of Stewart,
instead of arguing the question itself. I had asserted
that some axioms are necessary as the foundations of
mathematical reasoning, in addition to the definitions.
If Stewart did not intend to discuss this question, I had
no concern with what he had said about axioms. But I
had every reason to believe that this was the question
which Stewart did intend to discuss. I conceive there is
no doubt that he intended to give an opinion upon the
grounds of mathematical reasoning in general. For he
begins his discussions (Elements, Vol. n., p. 38) by contest
ing Reid's opinion on this subject, which is stated gene
rally; and he refers again to the same subject, asserting
in general terms, that the first principles of mathematics
are not axioms but definitions. If, then, afterwards, he
made his proof narrower than his assertion ; — if having
declared that no axioms are necessary, he afterwards
limited himself to showing that seven out of twelve of
Euclid's axioms arc barren truisms, it was no concern of
mine to contest this assertion, which left my thesis un
touched. I had asserted that the proper geometrical
axioms (that two straight lines cannot inclose a space,
and the axiom about parallel lines) are indispensable in
PHILOSOPHY OF THE PURE SCIENCES.
geometry. What account the reviewer gives of these
axioms we shall soon see; but if Stewart allowed them to
be axioms necessary to geometrical reasoning, he over
turned his own assertion as to the foundations of such
reasoning ; and if he said nothing decisive about these
axioms, which are the points on which the battle must
turn, he left his assertion altogether improved ; nor was
it necessary for me to pursue the war into a barren and
unimportant corner, when the metropolis was surrendered.
The reviewer's exultation that I have not contested the
first seven axioms is an amusing example of the self-
complacent zeal of advocacy.
3. But let us turn to the material point, — the proper
geometrical axioms. What is the reviewer's account of
these? Which side of the alternative does he adopt?
Do they depend upon the definitions, and is he prepared
to show the dependence ? Or are they superfluous, and
can he erect the structure of geometry without their aid?
One of these two courses, it would seem, he must take.
For we both begin by asserting the excellence of geo
metry as an example of demonstrated truth. It is
precisely this attribute which gives an interest to our
present inquiry. How, then, does the reviewer explain
this excellence on his views ? How does he reckon the
foundation courses of the edifice which we agree in con
sidering as a perfect example of intellectual building ?
I presume I may take, as his answer to this question,
his hypothetical statement of what Stewart would have
said, (p. 87,) on the supposition that there had been,
among the foundations of geometry, self-evident indemon
strable truths : although it is certainly strange that the
reviewer should not venture to make up his mind as to
the truth or falsehood of this supposition. If there were
such truths they would be, he says, " legitimate filiations"
of the definitions. They would be involved in the defi-
ANSWER TO OBJECTIONS. 1(>5
nitions. And again he speaks of the foundation of the
geometrical doctrine of parallels as a flaw, and as a
truth which requires, but has not received demonstration.
And yet again, he tells us that each of these supposed
axioms (Euclid's twelfth, for instance), is "merely an
indication of the point at which geometry fails to per
form that which it undertakes to perform" (p. 91); and
that in reality her truths are not yet demonstrated. The
amount of this is, that the geometrical axioms are to be
held to be legitimate filiations of the definitions, because
though certainly true, they cannot be proved from the
definitions ; that they are involved in the definitions,
although they cannot be evolved out of them ; and that
rather than admit that they have any other origin than
the definitions, we are to proclaim that geometry has
failed to perform what she undertakes to perform.
-\fo this I reply — that I cannot understand what is
meant by "legitimate filiations" of principles, if the phrase
not mean consequences of such principles established by
rigorous and formal demonstrations ; — that the reviewer,
if he claims any real signification for his phrase, must
substantiate the meaning of it by such a demonstration ;
he must establish his " legitimate filiation" by a genea
logical table in a satisfactory form. "When this cannot
be done, to assert, notwithstanding, that the propositions
are involved in the definitions, is a mere begging the
question; and to excuse this defect by saying that geo
metry fails to perform what she has promised, is to calum
niate the character of that science which we profess to
make our standard, rather than abandon an arbitrary
and unproved assertion respecting the real grounds of
her excellence. I add, further, that if the doctrine of
parallel lines, or any other geometrical doctrine of which
we see the truth, with the most perfect insight of its
necessity, have not hitherto received demonstration to the
106 PHILOSOPHY OF THE PURE SCIENCES.
satisfaction of any school of reasoners, the defect must
arise from their erroneous views of the nature of demon
strations, and the grounds of mathematical certainty.
4. I conceive, then, that the reviewer has failed alto
gether to disprove the doctrine that the axioms of geo
metry are necessary as a part of the foundations of the
science. I had asserted further that these axioms supply
what the definitions leave deficient ; and that they, along
with definitions, serve to present the idea of space under
such aspects that we can reason logically concerning it.
To this the reviewer opposes (p. 96) the common opinion
that a perfect definition is a complete explanation of a
name, and that the test of its perfection is, that we
may substitute the definition for the name wherever
it occurs. I reply, that my doctrine, that a definition
expresses a part, but not the whole, of the essential cha
racters of an idea, is certainly at variance with an opinion
sometimes maintained, that a definition merely explains
a word, and should explain it so fully that it may always
replace it. The error of this common opinion may, I think,
be shown from considerations such as these ; — that if wre
undertake to explain one word by several, we may be
called upon, on the same ground, to explain each of these
several by others, and that in this way we can reach no
limit nor resting-place ; — that in point of fact, it is not
found to lead to clearness, but to obscurity, when in the
discussion of general principles, we thus substitute defi
nitions for single terms ; — that even if this be done, we
cannot reason without conceiving what the terms mean ;
—and that, in doing this, the relations of our concep
tions, and not the arbitrary equivalence of two forms of
expression, are the foundations of our reasoning.
5. The reviewer conceives that some of the so-called
axioms are really definitions. The axiom, that " magni
tudes which coincide with each other, that is, which fill
ANSWER TO OBJECTIONS. 107
the same space, are equal," is a definition of geometrical
equality : the axiom, that " the whole is greater than its
part," is a definition of whole and part. But surely there
are very serious objections to this view. It would seem
more natural to say, if the former axiom is a definition
of the word equal, that the latter is a definition of the
word greater. And how can one short phrase define two
terms ? If I say, " the heat of summer is greater than
the heat of winter," does this assertion define anything,
though the proposition is perfectly intelligible and dis
tinct? I think, then, that this attempt to reduce these
axioms to definitions is quite untenable.
6. I have stated that a definition can be of no use,
except we can conceive the possibility and truth of the
property connected with it ; and that if we do conceive
this, we may rightly begin our reasonings by stating the
property as an axiom ; which Euclid does, in the case of
straight lines and of parallels. The reviewer inquires,
(p. 92,) whether I am prepared to extend this doctrine to
the case of circles, for which the reasoning is usually
rested upon the definition ; — whether I would replace this
definition by an axiom, asserting the possibility of such a
circle. To this I might reply, that it is not at all incum
bent upon me to assent to such a change ; for I have all
along stated that it is indifferent whether the fundamen
tal properties from which we reason be exhibited as defi
nitions or as axioms, provided their necessity be clearly
seen. But I am ready to declare that I think the form
of our geometry would be not at all the worse, if, instead
of the usual definition of a circle, — " that it is a figure
contained by one line, which is called the circumference,
and which is such, that all straight lines drawn from a
certain point within the circumference are equal to one
another," — we were to substitute an axiom and a defini
tion, as follows : —
108 PHILOSOPHY OF THE PURE SCIENCES.
Axiom. If a line be drawn so as to be at every point
equally distant from a certain point, this line will return
into itself, or will be one line including a space.
Definition. The space is called a circle, the line the
circumference, and the point the center.
And this being done, it would be true, as the reviewer
remarks, that geometry cannot stir one step without
resting on an axiom. And I do not at all hesitate to say,
that the above axiom, expressed or understood, is no less
necessary than the definition, and is tacitly assumed in
every proposition into which circles enter.
7. I have, I think, now disposed of the principal
objections which bear upon the proper axioms of geo
metry. The principles which are stated as the first seven
axioms of Euclid's Elements, need not, as I have said, be
here discussed. They are principles which refer, not to
Space in particular, but to Quantity in general : such,
for instance, as these ; " If equals be added to equals the
wholes are equal ;" — " If equals be taken from equals
the remainders are equal." But I will make an obser
vation or two upon them before I proceed.
Both Locke and Stewart have spoken of these axioms
as barren truisms : as propositions from which it is not
possible to deduce a single inference : and the reviewer
asserts that they are not first principles, but laws of
thought, (p. 88.) To this last expression I am willing
to assent ; but I would add, that not only these, but all
the principles which express the fundamental conditions
of our knowledge, may with equal propriety be termed
laws of thought ; for these principles depend upon our
ideas, and regulate the active operations of the mind, by
which coherence and connexion are given to its passive
impressions. But the assertion that no conclusions can
be drawn from simple axioms, or laws of human thought,
which regard quantity, is by no means true. The whole
ANSWER TO OBJECTIONS.
of arithmetic, — for instance, the rules for the multiplica- \
tion and division of large numbers, for finding a common
measure, and, in short, a vast body of theory respecting
numbers, — rests upon no other foundation than such
axioms as have been just noticed, that if equals be added
to equals the wholes will be equal. And even when
Locke's assertion, that from these axioms no truths can
be deduced, is modified by Stewart and the reviewer,
and limited to geometrical truths, it is hardly tenable
(although, in fact, it matters little to our argument
whether it is or no). For the greater part of the Seventh
Book of Euclid's Elements, (on Commensurable and In
commensurable Quantities,) and the Fifth Book, (on
Proportion,) depend upon these axioms, with the addi
tion only of the definition or axiom (for it may be stated
either way) which expresses the idea of proportionality
in numbers. So that the attempt to disprove the neces
sity and use of axioms, as principles of reasoning, fails
even when we take those instances which the opponents
consider as the more manifestly favourable to their
doctrine.
8. But perhaps the question may have already sug
gested itself to the reader's mind, of what use can it be
formally to state such principles as these, (for example,
that if equals be added to equals the wholes are equal,)
since, whether stated or no, they will be assumed in our
reasoning? And how can such principles be said to be
necessary, when our proof proceeds equally well without
any reference to them? And the answer is, that it is
precisely because these are the common principles of
reasoning, which we naturally employ without specially
contemplating them, that they require to be separated
from the other steps and formally stated, when we
analyze the demonstrations which we have obtained
In every mental process many principles are combined
110 PHILOSOPHY OF THE PURE SCIENCES.
and abbreviated, and thus in some measure concealed
and obscured. In analyzing these processes, the combi
nation must be resolved, and the abbreviation expanded,
and thus the appearance is presented of a pedantic and
superfluous formality. But that which is superfluous for
proof, is necessary for the analysis of proof. In order to
exhibit the conditions of demonstration distinctly, they
must be exhibited formally. In the same manner, in
demonstration we do not usually express every step in
the form of a syllogism, but we see the grounds of the
conclusiveness of a demonstration, by resolving it into
syllogisms. Neither axioms nor syllogisms are necessary
for conviction ; but they are necessary to display the
conditions under which conviction becomes inevitable.
The application of a single one of the axioms just spoken
of is so minute a step in the proof, that it appears pe
dantic to give it a marked place ; but the very essence
of demonstration consists in this, that it is composed of
an indissoluble succession of such minute steps. The
admirable circumstance is, that by the accumulation of
such apparently imperceptible advances, we can in the
end make so vast and so sure a progress. The com
pleteness of the analysis of our knowledge appears in the
smallness of the elements into which it is thus resolved.
The minuteness of any of these elements of truth, of
axioms for instance, does not prevent their being as
essential as others which are more obvious. And any
attempt to assume one kind of element only, when the
course of our analysis brings before us two or more
kinds, is altogether unphilosophical. Axioms and defi
nitions are the proximate constituent principles of our
demonstrations ; and the intimate bond which connects
together a definition and an axiom on the same subject
is not truly expressed by asserting the latter to be de
rived from the former. This bond of connexion exists
OF THE PERCEPTION OF SPACE. Ill
in the mind of the reasoner, in his conception of that to
which both definition and axiom refer, and consequently
in the general Fundamental Idea of which that concep
tion is a modification.
CHAPTER VI.
OF THE PERCEPTION OF SPACE.
1. ACCORDING to the views above explained, certain
of the impressions of our senses convey to us the per
ception of objects as existing in space ; inasmuch as by
the constitution of our minds we cannot receive those
impressions otherwise than in a certain form, involving
such a manner of existence. But the question deserves
to be asked, What are the impressions of sense by which
we thus become acquainted with space and its relations ?
And as we have seen that this idea of space implies an
act of the mind as well as an impression on the sense,
what manifestations do we find of this activity of the
mind, in our observation of the external world?
It is evident that sight and touch are the senses by
which the relations of space are perceived, principally or
entirely. It does not appear that an odour, or a feeling
of warmth or cold, would, independently of experience,
suggest to us the conception of a space surrounding us.
But when we see objects, we see that they are extended
and occupy space ; when we touch them, we feel that
they are in a space in which we also are. We have
before our eyes any object, for instance, a board covered
with geometrical diagrams ; and we distinctly perceive,
by vision, those lines of which the relations are the
subjects of our mathematical reasoning. Again, we see
before us a solid object, a cubical box for instance ; we
see that it is within reach ; we stretch out the hand and
11:2 PHILOSOPHY OK THE PURE SCIENCES.
perceive by the touch that it has sides, edges, corners,
which we had already perceived by vision.
2. Probably most persons do not generally appre
hend that there is any material difference in these two
cases ; — that there are any different acts of mind con
cerned in perceiving by sight a mathematical diagram
upon paper, and a solid cube lying on a table. Yet it is
not difficult to show that, in the latter case at least, the
perception of the shape of the object is not immediate.
A very little attention teaches us that there is an act of
judgment as well as a mere impression of sense requisite,
in order that we may see any solid object. For there is
no visible appearance which is inseparably connected
with solidity. If a picture of a cube be rightly drawn in
perspective and skilfully shaded, the impression upon the
sense is the same as if it were a real cube. The picture
may be mistaken for a solid object. But it is clear that,
in this case, the solidity is given to the object by an act
of mental judgment. All that is seen is outline and
shade, figures and colours on a flat board. The solid
angles and edges, the relation of the faces of the figure
by which they form a cube, are matters of inference.
This, which is evident in the case of the pictured cube, is
true in all vision whatever. We see a scene before us
on which are various figures and colours, but the eye
cannot see more. It sees length and bi'eadth, but no
third dimension. In order to know that there are solids,
we must infer as well as see. And this we do readily
and constantly ; so familiarly, indeed, that we do not
perceive the operation. Yet we may detect this latent
process in many ways; for instance, by attending to
cases in which the habit of drawing such inferences mis
leads us. Most persons have experienced this delusion
in looking at a scene in a theatre, and especially that
kind of scene which is called a diorama, when the
OF THE PERCEPTION OF SPACE. 1 1 3
interior of a building is represented. In these cases,
the perspective representations of the various members
of the architecture and decoration impress us almost
irresistibly with the conviction that we have before us a
space of great extent and complex form, instead of a flat
painted canvass. Here, at least, the space is our own
creation, but yet here, it is manifestly created by the
same act of thought as if we were really in the palace or
the cathedral of which the halls and aisles thus seem to
inclose us. And the act by which we thus create space
of three dimensions out of visible extent of length and
breadth, is constantly and imperceptibly going on. We
are perpetually interpreting in this manner the language
of the visible world. From the appearances of things
which we directly see, we are constantly inferring that
which we cannot directly see, — their distance from us,
and the position of their parts.
3. The characters which we thus interpret are
various. They are, for instance, the visible forms,
colours, and shades of the parts, understood according
to the maxims of perspective ; (for of perspective every
one has a practical knowledge, as every one has of
grammar ;) the effort by which we fix both our eyes on
the same object, and adjust each eye to distinct vision ;
and the like. The right interpretation of the informa
tion which such circumstances give us respecting the
true forms and distances of things, is gradually learned ;
the lesson being begun in our earliest infancy, and
inculcated upon us every hour during which we use our
eyes. The completeness with which the lesson is mas
tered is truly admirable; for we forget that our con
clusion is obtained indirectly, and mistake a judgment
on evidence for an intuitive perception. We see the
breadth of the street, as clearly and readily as we see
the house on the other side of it ; and we see the house
VOL. i. w. p. 1
114 PHILOSOPHY OF THE PURE SCIENCES.
to be square, however obliquely it be presented to us.
This, however, by no means throws any doubt or diffi
culty on the doctrine that in all these cases we do inter
pret and infer. The rapidity of the process, and the
unconsciousness of the effort, are not more remarkable
in this case than they are when we understand the
meaning of the speech which we hear, or of the book
which we read. In these latter cases we merely hear
noises or see black marks ; but we make, out of these
elements, thought and feeling, without being aware of
the act by which we do so. And by an exactly similar
process we see a variously-coloured expanse, and collect
from it a space occupied by solid objects. In both
cases the act of interpretation is become so habitual
that we can hardly stop short at the mere impression
of sense.
4. But yet there are various ways in which we may
satisfy ourselves that these two parts of the process of
seeing objects are distinct. To separate these operations
is precisely the task which the artist has to execute in
making a drawing of what he sees. He has to recover
the consciousness of his real and genuine sensations, and
to discern the lines of objects as they appear. This at
first he finds difficult ; for he is tempted to draw what
he knows of the forms of visible objects, and not what
he sees : but as he improves in his art, he learns to put
on paper what he sees only, separated from what he
infers, in order that thus the inference, and with it a
conception like that of the reality, may be left to the
-spectator. And thus the natural process of vision is the
habit of seeing that which cannot be seen ; and the diffi
culty of the .art of drawing consists in learning not to
see more than is visible.
5. But again ; even in the simplest drawing we
exhibit something which we do not see. However
OF THE PERCEPTION OF SPACE. 1 1 f)
slight is our representation of objects, it contains some
thing which we create for ourselves. For we draw an
outline. Now an outline has no existence in nature.
There are no visible lines presented to the eye by a
group of figures. We separate each figure from the rest,
and the boundary by which we do this is the outline of
the figure ; and the like may be said of each member of
every figure. A painter of our own times has made this
remark in a work upon his art*. "The effect which
natural objects produce upon our sense of vision is that
of a number of parts, or distinct masses of form and
colour, and not of lines. But when we endeavour to
represent by painting the objects which are before us, or
which invention supplies to our minds, the first and the
simplest means we resort to is this picture, by which we
separate the form of each object from those that sur
round it, marking its boundary, the extreme extent of
its dimensions in every direction, as impressed on our
vision : and this is termed drawing its outline."
6. Again, there are other ways in which we see clear
manifestations of the act of thought by which we assign
to the parts of objects their relations in space, the im
pressions of sense being merely subservient to this act.
If we look at a medal through a glass which inverts it,
we see the figures upon it become concave depressions
instead of projecting convexities ; for the light which
illuminates the nearer side of the convexity will be trans
ferred to the opposite side by the apparent inversion of
the medal, and will thus imply a hollow in which the
side nearest the light gathers the shade. Here our deci
sion as to which part is nearest to us, has reference to
the side from which the light comes. In other cases
the decision is more spontaneous. If we draw black
outlines, such as represent the edges of a cube seen
* Phillips On Painting.
I 2
11G PHILOSOPHY OF THE PURE SCIENCES.
in perspective, certain of the lines will cross each other ;
and we may make this cube appear to assume two dif
ferent positions, by determining in our own mind that
the lines which belong to one end of the cube shall be
understood to be before or to be behind those which
they cross. Here an act of the will, operating upon the
same sensible image, gives us two cubes, occupying two
entirely different positions. Again, many persons may
have observed that when a windmill in motion at a dis
tance from us, (so that the outline of the sails only is
seen,) stands obliquely to the eye, we may, by an effort
of thought, make the obliquity assume one or the other
of two positions ; and as we do this, the sails, which in
one instance appear to turn from right to left, in the other
case turn from left to right. A person a little familiar
with this mental effort, can invert the motion as often as
he pleases, so long as the conditions of form and light
do not offer a manifest contradiction to either position.
Thus we have these abundant and various manifesta
tions of the activity of the mind, in the process by which
we collect from vision the relations of solid space of three
dimensions. But we must further make some remarks
on the process by which we perceive mere visible figure;
and also, on the mode in which we perceive the relations
of space by the touch ; and first, of the latter subject.
7. The opinion above illustrated, that our sight does
not give us a direct knowledge of the relations of solid
space, and that this knowledge is acquired only by an
inference of the mind, was first clearly taught by the
celebrated Bishop Berkeley"", and is a doctrine now
generally assented to by metaphysical speculators.
But does the sense of touch give us directly a know
ledge of space ? This is a question which has attracted
considerable notice in recent times ; and new light has
* Theory of Vision.
OF THE PERCEPTION OF SPACE. 117
been thrown upon it in a degree which is very remark
able, when we consider that the philosophy of perception
has been a prominent subject of inquiry from the earliest
times. Two philosophers, advancing to this inquiry from
different sides, the one a metaphysician, the other a phy
siologist, have independently arrived at the conviction
that the long current opinion, according to which we
acquire a knowledge of space by the sense of touch, is
erroneous. And the doctrine which they teach instead
of the ancient errour, has a very important bearing upon
the principle which we are endeavouring to establish,—
that our knowledge of space and its properties is derived
rather from the active operations than from the passive
impressions of the percipient mind.
Undoubtedly the persuasion that we acquire a know
ledge of form by the touch is very obviously suggested
by our common habits. If we wish to know the form of
any body in the dark, or to correct the impressions con
veyed by sight, when we suspect them to be false, we
have only, it seems to us, at least at first, to stretch forth
the hand and touch the object ; and we learn its shape
with no chance of error. In these cases, form appears
to be as immediate a perception of the sense of touch,
as colour is of the sense of sight.
8. But is this perception really the result of the
passive sense of touch merely ? Against such an opinion
Dr. Brown, the metaphysician of whom I speak, urges*
that the feeling of touch alone, when any object is ap
plied to the hand, or any other part of the body, can no
more convey the conception of form or extension, than
the sensation of an odour or a taste can do, except we
have already some knowledge of the relative position of
the parts of our bodies; that is, except we are already in
possession of an idea of space, and have, in our minds,
* Lectures, Vol. i. p. 459, (182-1).
118 PHILOSOPHY OF THE PURE SCIENCES.
referred our limbs to their positions; which is to sup
pose the conception of form already acquired.
9. By what faculty then do we originally acquire our
conceptions of the relations of position ? Brown answers
by the muscular sense ; that is, by the conscious exer
tions of the various muscles by which we move our limbs.
When we feel out the form and position of bodies by
the hand, our knowledge is acquired, not by the mere
touch of the body, but by perceiving the course the
fingers must take in order to' follow the surface of the
body, or to pass from one body to another. We are
conscious of the slightest of the volitions by which we
thus feel out form and place ; we know whether we move
the finger to the right or left, up or down, to us or from
us, through a large or a small space ; and all these con
scious acts are bound together and regulated in our
minds by an idea, of an extended space in which they are
performed. That this idea of space is not borrowed from
the sight, and transferred to the muscular feelings by
habit, is evident. For a man born blind can feel out his
way with his staff, and has his conceptions of position
determined by the conditions of space, no less than one
who has the use of his eyes. And the muscular con
sciousness which reveals to us the position of objects and
parts of objects, when we feel them out by means of the
hand, shews itself in a thousand other ways, and in all
our limbs : for our habits of standing, walking, and all
other attitudes and motions, are regulated by our feeling
of our position and that of surrounding objects. And
thus, we cannot touch any object without learning some
thing respecting its position ; not that the sense of
touch directly conveys such knowledge ; but we have
already learnt, from the muscular sense, constantly
exercised, the position of the limb which the object thus
touches.
OF THE PERCEPTION OF SPACE. 1 1 9
10. The justice of this distinction will, I think, be
assented to by all persons who attend steadily to the
process itself, and might be maintained by many forcible
reasons. Perhaps one of the most striking evidences in
its favour is that, as I have already intimated, it is the
opinion to which another distinguished philosopher, Sir
Charles Bell, has been led, reasoning entirely upon phy
siological principles. From his researches it resulted
that besides the nerves which convey the impulse of the
will from the brain to the muscle, by which every motion
of our limbs is produced, there is another set of nerves
which carry back to the brain a sense of the condition
of the muscle, and thus regulate its activity ; and give us
the consciousness of our position and relation to sur
rounding objects. The motion of the hand and fingers,
or the consciousness of this motion, must be combined
with the sense of touch properly so called, in order to
make an inlet to the knowledge of such relations. This
consciousness of muscular exertion, which he has called a
sixth sense"", is our guide, Sir C. Bell shows, in the com
mon practical government of our motions ; and he states
that having given this explanation of perception as a
physiological doctrine, he had afterwards with satisfac
tion seen it confirmed by Dr. Brown's speculations.
11. Thus it appears that our consciousness of the
relations of space is inseparably and fundamentally con
nected with our own actions in space. We perceive only
while we act ; our sensations require to be interpreted by
our volitions. The apprehension of extension and figure
is far from being a process in which we are inert and
passive. We draw lines with our fingers ; we construct
surfaces by curving our hands; we generate spaces by the
motion of our arms. When the geometer bids us form
lines, or surfaces, or solids by motion, he intends his
* Bridgetvater Treatise, p. 195. Phil. Trans. 1K2<>, I't. 11., p. H>7-
120 PHILOSOPHY OF THE PURE SCIENCES.
injunction to be taken as hypothetical only ; we need only
conceive such motions. But yet this hypothesis repre
sents truly the origin of our knowledge ; we perceive
spaces by motion at first, as we conceive spaces by motion
afterwards : — or if not always by actual motion, at least
by potential. If we perceive the length of a staff by
holding its two ends in our two hands without running
the finger along it, this is because by habitual motion we
have already acquired a measure of the distance of our
hands in any attitude of which we are conscious. Even
in the simplest case, our perceptions are derived not from
the touch, but from the sixth sense ; and this sixth sense
at least, whatever may be the case with the other five,
implies an active mind along with the passive sense.
12. Upon attentive consideration, it will be clear
that a large portion of the perceptions respecting space
which appear at first to be obtained by sight alone, are,
in fact, acquired by means of this sixth sense. Thus we
consider the visible sky as a single surface surrounding
us and returning into itself, and thus forming a hemi
sphere. But such a mode of conceiving an object of vision
could never have occurred to us, if we had not been able
to turn our heads, to follow this surface, to pursue it till
we find it returning into itself. And when we have done
this, we necessarily present it to ourselves as a concave
inclosure within which we are. The sense of sight alone,
without the power of muscular motion, could not have
led us to view the sky as a vault or hemisphere. Under
such circumstances, we should have perceived only what
was presented to the eye in one position ; and if dif
ferent appearances had been presented in succession, we
could not have connected them as parts of the same
picture, for want of any perception of their relative posi
tion. They would have been so many detached and
incoherent visual sensations. The muscular sense con-
OF THE PERCEPTION OF SPACE. 121
nccts their parts into a whole, making them to be only
different portions of one universal scene *.
13. These considerations point out the fallacy of a
very curious representation made by Dr. Reid, of the
convictions to which man would be led, if he possessed
vision without the sense of touch. To illustrate this sub
ject, Reid uses the fiction of a nation whom he terms the
Idomenians, who have no sense except that of sight. He
describes their notions of the relations of space as being
entirely different from ours. The axioms of their geome
try are quite contradictory to our axioms. For example,
it is held to be self-evident among them that two straight
lines which intersect each other once, must intersect a
second time ; that the three angles of any triangle are
greater than two right angles; and the like. These
paradoxes are obtained by tracing the relations of lines
on the surface of a concave sphere, which surrounds the
spectator, and on which all visible appearances may be
supposed to be presented to him. But from what is said
above it appears that the notion of such a sphere, and
such a connexion of visible objects which are seen in dif
ferent directions, cannot be arrived at by sight alone.
* It has been objected to this view, that we might obtain a con
ception of the sky as a hemisphere, by being ourselves turned round, (as
on a music-stool, for instance,) and thus seeing in succession all parts of
the sky. But this assertion I conceive to be erroneous. By being thus
turned round, we should see a number of pictures which we should put
together as parts of a plane picture ; and when we came round to the
original point, we should have no possible means of deciding that it
mas the same point : it would appear only as a repetition of the pic
ture. That sight, of itself, can give us only a plane picture, the doctrine
of Berkeley, appears to be indisputable ; and, no less so, the doctrine
that it is the consciousness of our own action in space which puts toge
ther these pictures so that they cover the surface of a solid body. We
can see length and breadth with our eyes, but we must thrust out our
arm towards the flat surface, in order that we may, in our thoughts,
combine a third dimension with the other two.
122 PHILOSOPHY OF THE PURE SCIENCES.
When the spectator combines in his conception the rela
tions of long-drawn lines and large figures, as he sees
them by turning his head to the right and to the left,
upwards and downwards, he ceases to be an Idomenian.
And thus our conceptions of the properties of space, de
rived through the exercise of one mode of perception,
are not at variance with those obtained in another way ;
but all such conceptions, however produced or suggested,
are in harmony with each other ; being, as has already
been said, only different aspects of the same idea.
14. If our perceptions of the position of objects
around us do not depend on the sense of vision alone,
but on the muscular feeling brought into play when we
turn our head, it will obviously follow that the same is
true when we turn the eye instead of the head. And
thus we may learn the form of objects, not by looking
at them with a fixed gaze, but by following the boundary
of them with the eye. While the head is held perfectly
still, the eye can rove along the outlines of visible ob
jects, scrutinize each point in succession, and leap from
one point to another ; each such act being accompanied
by a muscular consciousness which makes us aware of
the direction in which the look is travelling. And we
may thus gather information concerning the figures and
places which we trace out with the visual ray, as the
blind man learns the forms of things which he traces out
with his staff, being conscious of the motions of his hand.
15. This view of the mode in which the eye per
ceives position, which is thus supported by the analogy
of other members employed for the same purpose, is
further confirmed by Sir Charles Bell by physiological
reasons. He teaches us that""" when an object is seen we
employ two senses: there is an impression on the retina;
but we receive also the idea of position or relation in
* Phil. Trans., 1823. On the Motions of the Eye.
OF THE PERCEPTION OF SPACE. 1 ±3
space, which it is not the office of the retina to give, by
our consciousness of the efforts of the voluntary muscles
of the eve : and he has traced in detail the course of the
•/
nerves by which these muscles convey their information.
The constant searching motion of the eye, as he terms
it*, is the means by which we become aware of the
position of objects about us.
16. It is not to our present purpose to follow' the
physiology of this subject ; but we may notice that Sir
C. Bell has examined the special circumstances which
belong to this operation of the eye. We learn from him
that the particular point of the eye which thus traces the
forms of visible objects is a part of the retina which has
been termed the sensible spot; being that part which is
most distinctly sensible to the impressions of light and
colour. This part, indeed, is not a spot of definite size and
form, for it appears that proceeding from a certain point
of the retina, the distinct sensibility diminishes on every
side by degrees. And the searching motion of the eye
arises from the desire which we instinctively feel of re
ceiving upon the sensible spot the image of the object
to which the attention is directed. We are uneasy and
* Bridgtwater Treatise, p. 282. I have adopted, in writing the
ahove, the views and expressions of Sir Charles Bell. The essential
part of the doctrine there presented is, that the eye constantly makes
efforts to turn, so that the image of an object to which our attention is
drawn, shall fall upon a certain particular point of the retina; and that
when the image falls upon any other point, the eye turns away from
this oblique into the direct position. Other writers have maintained
that the eye thus turns, not because the point on which the image falls
in direct vision is the moxt .sensible point, but that it is the point of
greatest distinctness of vision. They urge that a small star, which dis
appears when the eye is turned full upon it, may often be seen by
looking a little away from it : and hence, they infer that the parts of
the retina removed from the spot of direct vision, are more sensible than
it is. The facts are very curious, however they lx) explained, but they
do not disturb the doetrinc delivered in the text.
124 PHILOSOPHY OF THE PURE SCIENCES.
impatient till the eye is turned so that this is effected.
And as our attention is transferred from point to point
of the scene before us, the eye, and this point of the eye
in particular, travel along with the thoughts ; and the
muscular sense, which tells us of these movements of
the organ of vision, conveys to us a knowledge of the
forms and places which we thus successively survey.
17. How much of activity there is in the process by
which we perceive the outlines of objects appears further
from the language by which we describe their forms.
We apply to them not merely adjectives of form, but
verbs of motion. An abrupt hill starts out of the plain ;
a beautiful figure has a gliding outline. We have
The windy summit, wild and high,
Roughly rushing on the sky.
These terms express the course of the eye as it follows
the lines by which such forms are bounded and marked.
In like manner another modern poet* says of Soracte,
that it
From out the plain
Heaves like a long-swept wave about to break,
And on the curl hangs pausing.
Thus the muscular sense, which is inseparably con
nected with an act originating in our own mind, not only
gives us all that portion of our perceptions of space in
which wre use the sense of touch, but also, at least in a
great measure, another large portion of such perceptions,
in which we employ the sense of sight. As we have
before seen that our knowledge of solid space and its
properties is not conceivable in any other way than as
the result of a mental act, governed by conditions depend
ing on its own nature ; so it nowT appears that our per
ceptions of visible figure are not obtained without an act
performed under the same conditions. The sensations
of touch and sight are subordinated to an idea which is
* Byron, Ch. Har. vi., st. 75.
OF THE PERCEPTION OF SPACE. \'2i)
the basis of our speculative knowledge concerning space
and its relations ; and this same idea is disclosed to our
consciousness by its practically regulating our inter
course with the external world.
By considerations such as have been adduced and
referred to, it is proved beyond doubt, that in a great
number of cases our knowledge of form and position is
acquired from the muscular sense, and not from sight
directly: — for instance, in all cases in which we have
before us objects so large and prospects so extensive
that we cannot see the whole of them in one position of
the eye*.
We now quit the consideration of the properties of
Space, and consider the Idea of Time.
CHAPTER VII.
OF THE IDEA OF TIME.
1. RESPECTING the Idea of Time, we may make
several of the same remarks which we made concerning
* The expression in the first edition was " large objects and exten
sive spaces." In the text as now given, I state a definite size and
extent, within which the sight by itself can judge of position and figure.
The doctrine that we require the assistance of the muscular sense to
enable us to perceive space of three dimensions, is not at all inconsistent
with this other doctrine, that within the space which is seen by the
fixed eye, we perceive the relative positions of points directly by vision,
and that, consequently, we have a perception of visible .figure.
Sir Charles Bell has said, (Phil. Trans. 1823, p. 181,) "It appears
to me that the utmost ingenuity will be at a loss to devise an explana
tion of that power by which the eye becomes acquainted with the
position and relation of objects, if the sense of muscular activity be
excluded which accompanies the motion of the eyeball." But surely we
should have no difficulty in perceiving the relation of the sides and
angles of a small triangle, placed before the eye, even if the muscles of
the eyeball were severed. This subject is resumed 13. iv. c. ii. sect. 1 1.
126 PHILOSOPHY OF THE PURE SCIENCES.
the idea of space, in order to shew that it is not bor
rowed from experience, but is a bond of connexion
among the impressions of sense, derived from a peculiar
activity of the mind, and forming a foundation both of
our experience and of our speculative knowledge.
Time is not a notion obtained by experience. Expe
rience, that is, the impressions of sense and our con
sciousness of our thoughts, gives us various percep
tions; and different successive perceptions considered
together exemplify the notion of change. But this very
connexion of different perceptions, — this successiveness,
—presupposes that the perceptions exist in time. That
things happen either together, or one after the other, is
intelligible only by assuming time as the condition under
which they are presented to us.
Thus time is a necessary condition in the presentation
of all occurrences to our minds. We cannot conceive
this condition to be taken away. We can conceive
time to go on while nothing happens in it ; but we can
not conceive anything to happen while time does not
go on.
It is clear from this that time is not an impression
derived from experience, in the same manner in which
we derive from experience our information concerning
the objects which exist, and the occurrences which take
place in time. The objects of experience can easily be
conceived to be, or not to be : — to be absent as well as
present. Time always is, and always is present, and
even in our thoughts we cannot form the contrary sup
position.
2. Thus time is something distinct from the matter
or substance of our experience, and may be considered
as a necessary form which that matter (the experience of
change) must assume, in order to be an object of con
templation to the mind. Time is one of the necessary
OF THE IDEA OF TIME. \'27
conditions under which we apprehend the information
which our senses and consciousness give us. By con
sidering time as a form which belongs to our power of
apprehending occurrences and changes, and under which
alone all such experience can be accepted by the mind,
we explain the necessity, which we find to exist, of con
ceiving all such changes as happening in time ; and we
thus see that time is not a property perceived as existing
in objects, or as conveyed to us by our senses ; but a con
dition impressed upon our knowledge by the constitution
of the mind itself; involving an act of thought as well as
an impression of sense.
3. We showed that space is an idea of the mind, or
form of our perceiving power, independent of experience,
by pointing out that we possess necessary and universal
truths concerning the relations of space, which could
never be given by means of experience ; but of which
the necessity is readily conceivable, if we suppose them
to have for their basis the constitution of the mind.
There exist also respecting number, many truths abso
lutely necessary, entirely independent of experience and
anterior to it ; and so far as the conception of number
depends upon the idea of time, the same argument might
be used to show that the idea of time is not derived from
experience, but is a result of the native activity of the
mind : but we shall defer all views of this kind till we
come to the consideration of Number.
4. Some persons have supposed that we obtain the
notion of time from the perception of motion. But it
is clear that the perception of motion, that is, change of
place, presupposes the conception of time, and is not
capable of being presented to the mind in any other way.
If we contemplate the same body as being in different
places at different times, and connect these observations,
we have the conception of motion, which thus presup-
128 PHILOSOPHY OF THE PURE SCIENCES.
poses the necessary conditions that existence in time
implies. And thus we see that it is possible there should
be necessary truths concerning all motion, and conse
quently, concerning those motions which are the objects
of experience ; but that the source of this necessity is the
Ideas of time and space, which, being universal conditions
of knowledge residing in the mind, afford a foundation
for necessary truths.
CHAPTER VIII.
OF SOME PECULIARITIES OF THE IDEA OF TIME.
1. THE Idea of Time, like the Idea of Space, offers to
our notice some characters which do not belong to our
fundamental ideas generally, but which are deserving of
remark. These characters are, in some respects, closely
similar with regard to time and to space, while, in other
respects, the peculiarities of these two ideas are widely
different. We shall point out some of these characters.
Time is not a general abstract notion collected from
experience ; as, for example, a certain general concep
tion of the relations of things. For we do not consider
particular times as examples of Time in general, (as we
consider particular causes to be examples of Cause,) but
we conceive all particular times to be parts of a single
and endless Time. This continually-flowing and endless
time is what offers itself to us when we contemplate any
series of occurrences. All actual and possible times
exist as Parts, in this original and general Time. And
since all particular times are considered as derivable
from time in general, it is manifest that the notion of
time in general cannot be derived from the notions of
particular times. The notion of time in general is there-
SOME PECULIARITIES OF THE IDEA OF TIME. 120
fore not a general conception gathered from experi
ence.
2. Time is infinite. Since all actual and possible
times exist in the general course of time, this general
time must be infinite. All limitation merely divides,
and does not terminate, the extent of absolute time.
Time has no beginning and no end ; but the beginning
and the end of every other existence takes place in it.
3. Time, like space, is not only a form of perception,
but of intuition. We contemplate events as taking
place in time. We consider its parts as added to one
another, and events as filling a larger or smaller extent
of such parts. The time which any event takes up is
the sum of all such parts, and the relation of the same
to time is fully understood when we can clearly see what
portions of time it occupies, and what it does not.
Thus the relation of known occurrences to time is
perceived by intuition ; and time is a form of intuition
of the external world.
4. Time is conceived as a quantity of one dimension ;
it has great analogy with a line, but none at all with a
surface or solid. Time may be considered as consisting
of a series of instants, which are before and after one
another; and they have no other relation than this, of
before and after. Just the same would be the case with
a series of points taken along a line ; each would be
after those on one side of it, and before those on another.
Indeed the analogy between time, and space of one
dimension, is so close, that the same terms are applied to
both ideas, and we hardly know to which they originally
belong. Times and lines are alike called long and short ;
we speak of the beginning and end of a line ; of a point
of time, and of the limits of a portion of duration.
5. But, as has been said, there is nothing in time
which corresponds to more than one dimension in space,
VOL. i. \v. r. K
130 PHILOSOPHY OF THE PURE SCIENCES.
and hence nothing which has any obvious analogy with
figure Time resembles a line indefinitely extended both
ways ; all partial times are portions of this line ; and no
mode of conceiving time suggests to us a line making
any angle with the original line, or any other combina
tion which might give rise to figures of any kind. The
analogy between time and space, which in many circum
stances is so clear, here disappears altogether. Spaces
of two and of three dimensions, planes and solids, have
nothing to which we can compare them in the concep
tions arising out of time.
6. As figure is a conception solely appropriate to
space, there is also a conception which peculiarly belongs
to time, namely, the conception of recurrence of times
similarly marked ; or, as it may be termed, rhythm,
using this word in a general sense. The term rhythm
is most commonly used to designate the recurrence of
times marked by the syllables of a verse, or the notes of
a melody : but it is easy to see that the general concep
tion of such a recurrence does not depend on the mode
in which it is impressed upon the sense. The forms of
such recurrence are innumerable. Thus in such a line as
Quadnipedante putrem sonitu quatit lingula campum,
we have alternately one long or forcible syllable, and
two short or light ones, recurring over and over. In
like manner in our own language, in the line
At the close of the day when the hamlet is still,
we have two light and one strong syllable repeated four
times over. Such repetition is the essence of versification.
The same kind of rhythm is one of the main elements of
music, with this difference only, that in music the forcible
syllables are made so for the purposes of rhythm by
their length only or principally ; for example, if either of
the above lines were imitated by a melody in the most
SOME PECULIARITIES OF THE IDEA OF TIME. 131
simple and obvious manner, each strong syllable would
occupy exactly twice as much time as two of the weaker
ones. Something very analogous to such rhythm may
be traced in other parts of poetry and art, which we need
not here dwell upon. But in reference to our present
subject, we may remark that by the introduction of such
rhythm, the flow of time, which appears otherwise so
perfectly simple and homogeneous, admits of an infinite
number of varied yet regular modes of progress. All
the kinds of versification which occur in all languages,
and the still more varied forms of recurrence of notes of
different lengths, which are heard in all the varied strains
of melodies, are only examples of such modifications, or
configurations as we may call them, of time. They in
volve relations of various portions of time, as figures
involve relations of various portions of space. But yet
the analogy between rhythm and figure is by no means
very close ; for in rhythm we have relations of quantity
alone in the parts of time, whereas in figure we have re
lations not only of quantity, but of a kind altogether
different, — namely, of position. On the other hand, a
repetition of similar elements, which does not necessarily
occur in figures, is quite essential in order to impress
upon us that measured progress of time of which we here
speak. And thus the ideas of time and space have each
its peculiar and exclusive relations ; position and figure
belonging only to space, while repetition and rhythm are
appropriate to time.
7. One of the simplest forms of recurrence is alter
nation, as when we have alternate strong and slight syl
lables. For instance,—
Awake, arise, or be for £ver fdll'n.
Or without any subordination, as when we reckon
numbers, and call them in succession, odd, even,
even.
K a
PHILOSOPHY OF THE PURE SCIENCES.
8. But the simplest of all forms of recurrence is that
which has no variety ; — in which a series of units, each
considered as exactly similar to the rest, succeed each
other ; as one, one, one, and so on. In this case, how
ever, we are led to consider each unit with reference to
all that have preceded ; and thus the series one, one, one,
and so forth, becomes one, two, three, four, Jive, and so
on ; a series with which all are familiar, and which may
be continued without limit.
We thus collect from that repetition of which time
admits, the conception of Number.
9. The relations of position and figure are the sub
ject of the science of geometry ; and are, as we have
already said, traced into a very remarkable and extensive
body of truths, which rests for its foundations on axioms
involved in the Idea of Space. There is, in like manner,
a science of great complexity and extent, which has its
foundation in the Idea of Time. But this science, as it
is usually pursued, applies only to the conception of Num
ber, which is, as we have said, the simplest result of
repetition. This science is Theoretical Arithmetic, or
the speculative doctrine of the properties and relations
of numbers ; and we must say a few words concerning
the principles which it is requisite to assume as the basis
of this science.
CHAPTER IX.
OF THE AXIOMS WHICH RELATE TO NUMBER.
I. TPIE foundations of our speculative knowledge of
the relations and properties of Number, as well as of
Space, are contained in the mode in which we represent to
ourselves the magnitudes which are the subjects of our
reasonings. To express these foundations in axioms in the
OF THE AXIOMS WHICH RELATE TO NUMBKK. 133
case of number, is a matter requiring some consideration,
for the same reason as in the case of geometry ; that is,
because these axioms are principles which we assume as
true, without being aware that we have made any assump
tion ; and we cannot, without careful scrutiny, determine
when we have stated, in the form of axioms, all that is
necessary for the formation of the science, and no more
than is necessary. We will, however, attempt to detect
the principles which really must form the basis of theo
retical arithmetic.
2. Why is it that three and two are equal to four and
one? Because if we look at five things of any kind, we
see that it is so. The five are four and one ; they are
also three and two. The truth of our assertion is in
volved in our being able to conceive the number five at
all. We perceive this truth by intuition, for we cannot
see, or imagine we see, five things, without perceiving
also that the assertion above stated is true.
But how do we state in words this fundamental prin
ciple of the doctrine of numbers ? Let us consider a
very simple case. If we wish to show that seven and
two are equal to four and five, we say that seven are four
and three, therefore seven and two are four and three
and two ; and because three and two are five, this is four
and five. Mathematical reasoners justify the first infer
ence (marked by the conjunctive word therefore), by
saying that "When equals are added to equals the
wholes are equal," and that thus, since seven is equal
to three and four, if we add two to both, seven and two
are equal to four and three and two.
3. Such axioms as this, that when equals are added
to equals the wholes are equal, are, in fact, expressions
of the general condition of intuition, by which a whole
is contemplated as made up of parts, and as identical
with the aggregate of the parts. And a yet more
1:34 PHILOSOPHY OF THE PURE SCIENCES.
ral form in which we might more adequately express
this conditon of intuition would be this ; that " Two mag
nitudes are equal when they can be divided into parts
which are equal, each to each." Thus in the above ex
ample, seven and two are equal to four and five, because
each of the two sums can be divided into the parts, four,
three, and two.
4. In all these cases, a person who had never seen
such axioms enunciated in a verbal form would employ
the same reasoning as a practised mathematician, in order
to satisfy himself that the proposition was true. The
steps of the reasoning, being seen to be true by intuition,
would carry an entire conviction, whether or not the
argument were made verbally complete. Hence the
axioms may appear superfluous, and on this account
such axioms have often been spoken contemptuously of
as empty and barren assertions. In fact, however, al
though they cannot supply the deficiency of the clear in
tuition of number and space in the reasoner himself, and
although when he possesses such a faculty, he will reason
rightly if he have never heard of such axioms, they still
have their place properly at the beginning of our trea
tises on the science of quantity ; since they express, as
simply as words can express, those conditions of the
intuition of magnitudes on which all reasoning concern
ing quantity must be based ; and are necessary when we
want, not only to see the truth of the elementary reason
ings on these subjects, but to put such reasonings in a
formal and logical shape.
5. We have considered the above-mentioned axioms
as the basis of all arithmetical operations of the nature
of addition. But it is easily seen that the same prin
ciple may be carried into other cases ; as for instance,
multiplication, which is merely a repeated addition,
and admits of the same kind of evidence. Thus
OF THE AXIOMS WHICH RELATE TO NUMIJEK. 135
five times three are equal to three times five ; why
is this ? If we arrange fifteen things in five rows of
three, it is seen by looking, or by imaginary looking,
which is intuition, that they may also be taken as three
rows of five. And thus the principle that those wholes
are equal which can be resolved into the same partial
magnitudes, is immediately applicable in this as in the
other case.
C. We may proceed to higher numbers, and may find
ourselves obliged to use artificial nomenclature and
notation in order to represent and reckon them ; but the
reasoning in these cases also is still the same. And the
usual artifice by which our reasoning in such instances
is assisted is, that the number which is the root of our
scale of notation (which is ten in our usual system), is
alternately separated into parts and treated as a single
thing. Thus 47 and 35 are 82 ; for 47 is four tens and
seven ; 35 is three tens and five ; whence 47 and 35 are
seven tens and twelve ; that is, 7 tens, 1 ten, and 2 ;
which is 8 tens and 2, or 82. The like reasoning is
applicable in other cases. And since the most remote
and complex properties of numbers are obtained by a
prolongation of a course of reasoning exactly similar to
that by which we thus establish the most elementary
propositions, we have, in the principles just noticed, the
foundation of the whole of Theoretical Arithmetic.
CHAPTER X.
OF THE PERCEPTION OF TIME AND NUMBER.
1. OUR perception of the passage of time involves a
series of acts of memory. This is easily seen and assented
to, when large intervals of time and a complex train of
occurrences are concerned. But since memory is requi-
130 PHILOSOPHY OF THE PURE SCIENCES.
site iii order to apprehend time in such cases, we cannot
doubt that the same faculty must be concerned in the
shortest and simplest cases of succession ; for it will
hardly be maintained that the process by which we con
template the progress of time is different when small
and when large intervals are concerned. If memory be
absolutely requisite to connect two events which begin
and end a day, and to perceive a tract of time between
them, it must be equally indispensable to connect the
beginning and end of a minute, or a second ; though in
this case the effort may be smaller, and consequently
more easily overlooked. In common cases, we are un
conscious of the act of thought by which we recollect
the preceding instant, though we perceive the effort when
we recollect some distant event. And this is analogous
to what happens in other instances. Thus, we walk
without being conscious of the volitions by which we
move our muscles ; but, in order to leap, a distinct and
manifest exertion of the same muscles is necessary. Yet
no one will doubt that we walk as well as leap by an
act of the will exerted through the muscles ; and in like
manner, our consciousness of small as well as large inter
vals of time involves something of the nature of an act
of memory.
2. But this constant and almost imperceptible kind
of memory, by which we connect the beginning and end
of each instant as it passes, may very fitly be distinguished
in common cases from manifest acts of recollection,
although it may be difficult or impossible to separate
the two operations in general. This perpetual and latent
kind of memory may be termed a sense of successive
ness ; and must be considered as an internal sense by
which we perceive ourselves existing in time, much in
the same way as by our external and muscular sense
we perceive ourselves existing in space. And both our
PERCEPTION OF TIME AND NUMBER. 137
internal thoughts and feelings, and the events which
take place around us, are apprehended as objects of this
internal sense, and thus as taking place in time.
3. In the same manner in which our interpretation
of the notices of the muscular sense implies the power of
moving our limbs, and of touching at will this object or
that ; our apprehension of the relations of time by means
of the internal sense of successiveness implies a power of
recalling what has past, and of retaining what is pass
ing. We are able to seize the occurrences which have
just taken place, and to hold them fast in our minds
so as mentally to measure their distance in time from
occurrences now present. And thus, this sense of suc
cessiveness, like the muscular sense with which we have
compared it, implies activity of the mind itself, and is
not a sense passively receiving impressions.
4. The conception of Number appears to require the
exercise of the same sense of succession. At first sight,
indeed, we seem to apprehend Number without any act
of memory, or any reference to time : for example, we
look at a horse, and see that his legs are four ; and this
we seem to do at once, without reckoning them. But it
is not difficult to see that this seeming instantaneousness
of the perception of small numbers is an illusion. This
resembles the many other cases in which we perform
short and easy acts so rapidly and familiarly that we are
unconscious of them; as in the acts of seeing, and of arti
culating our words. And this is the more manifest, since
we begin our acquaintance with number by counting
even the smallest numbers. Children and very rude
savages must use an effort to reckon even their five
fingers, and find a difficulty in going further. And per
sons have been known who were able by habit, or by a
peculiar natural aptitude, to count by dozens as rapidly
as common persons can by units. We may conclude,
138 PHILOSOPHY OF THE PURE SCIENCES.
therefore, that when we appear to catch a small number
by a single glance of the eye, we do in fact count the
units of it in a regular, though very brief succession. To
count requires an act of memory. Of this we are sen
sible when we count very slowly, as when we reckon the
strokes of a church-clock ; for in such a case we may
forget in the intervals of the strokes, and miscount. Now
it will not be doubted that the nature of the process in
counting is the same whether we count fast or slow.
There is no definite speed of reckoning at which the
faculties which it requires are changed; and therefore
memory, which is requisite in some cases, must be so
in all*.
The act of counting, (one, two, three, and so on,) is
the foundation of all our knowledge of number. The
intuition of the relations of number involves this act of
counting; for, as we have just seen, the conception of
number cannot be obtained in any other way. And thus
the whole of theoretical arithmetic depends upon an act
of the mind, and upon the conditions which the exercise
of that act implies. These have been already explained
in the last chapter.
5. But if the apprehension of number be accompanied
by an act of the mind, the apprehension of rhythm is so
still more clearly. All the forms of versification and the
measures of melodies are the creations of man, who thus
realizes in words and sounds the forms of recurrence
which rise within his own mind. When we hear in a
* I have considered Number as involving the exercise of the sense
of succession, because I cannot draw any line between those cases of
large numbers, in which, the process of counting being performed, there
is a manifest apprehension of succession ; and those cases of small num
bers, in which we seem to see the number at one glance. But if any
one holds Number to be apprehended by a direct act of intuition, as
Space and Time are, this view will not disturb the other doctrines
delivered .in the text.
PERCEPTION OF TIME AND NUMBER. 131)
quiet scene any rapidly-repeated sound, as those made by
the hammer of the smith or the saw of the carpenter,
every one knows how insensibly we throw these noises
into a rhythmical form in our own apprehension. We
do this even without any suggestion from the sounds
themselves. For instance, if the beats of a clock or
watch be ever so exactly alike, we still reckon them
alternately tick-tack, tick-tack. That this is the case,
may be proved by taking a watch or clock of such a con
struction that the returning swing of the pendulum is
silent, and in which therefore all the beats are rigorously
alike : we shall find ourselves still reckoning its sounds
as tick-tack. In this instance it is manifest that the
rhythm is entirely of our own making. In melodies,
also, and in verses in which the rhythm is complex, ob
scure, and difficult, we perceive something is required
on our part ; for we are often incapable of contributing
our share, and thus lose the sense of the measure alto
gether. And when we consider such cases, and attend
to what passes within us when we catch the measure,
even of the simplest and best-known air, we shall no
longer doubt that an act of our own thoughts is requisite
in such cases, as well as impressions on the sense. And
thus the conception of this peculiar modification of time,
which we have called rhythm, like all the other views
which we have taken of the subject, shows that we must,
in order to form such conceptions, supply a certain idea
by our own thoughts, as well as merely receive by senses,
whether external or internal, the impressions of appear
ances and collections of appearances.
NOTE TO CHAPTER X.
I HAVE in the last ten chapters described Space, Time, and Xunilwr by
various expressions, all intended to point out their office as exemplifying
the Ideal Element of human knowledge. I have called them Funda-
140 PHILOSOPHY OF THE PURE SCIENCES.
mental Ideas ; Forms of Perception ; Forms of Intuition ; and per
haps other names. I might add yet other phrases. I might say that
the properties of Space, Time, and Number are Laws of the Mind's
Activity in apprehending what is. For the mind cannot apprehend any
thing or event except conformably to the properties of space, time, and
number. It is not only that it does not, but it can not : and this
impossibility shows that the law is a law of the mind, and not of
objects extraneous to the mind.
It is usual for some of those who reject the doctrines here presented
to say that the axioms of geometry, and of other sciences, are obtained
by Induction from facts constantly presented by experience. But I do
not see how Induction can prove that a proposition must be true. The
only intelligible usage of the word Induction appears to me to be, that in
which it is applied to a proposition which, being separable from the
facts in our apprehension, and being compared with them, is seen to
agree with them. But in the cases now spoken of, the proposition is
not separable from the facts. We cannot infer by induction that two
straight lines cannot inclose a space, because we cannot contemplate
special cases of two lines inclosing a space, in which it remains to be
determined whether or not the proposition, that both are straight,
is true.
I do not deny that the activity of the mind by which it perceives
objects and events as related according to the laws of space, time, and
number, is awakened and developed by being constantly exercised ; and
that we cannot imagine a stage of human existence in which the powers
have not been awakened and developed by such exercise. In this way,
experience and observation are necessary conditions and prerequisites of
our apprehension of geometrical (and other) axioms. We cannot see
the truth of these axioms without some experience, because we cannot
see any thing, or be human beings, without some experience. This
might be expressed by saying that such truths are acquired necessarily
in the course of all experience ; but I think it is very undesirable to
apply, to such a case, the word Induction, of which it is so important
to us to keep the scientific meaning free from confusion. Induction
cannot give demonstrative proofs, as I have already stated in Book i.
C. ii. sect. 3, and therefore cannot be the ground of necessary truths.
Another expression which may be used to describe the Funda
mental Ideas here spoken of is suggested by the language of a very
profound and acute Review of the former edition. The Reviewer holds
that we pass from special experiences to universal truths in virtue of
*' the inductive propensity — the irresistible impulse of the mind to
generalize ad in/initnm." I have already given reasons why I cannot
adopt the former expression ; but I do not see why space, time, number,
PERCEPTION OF TIME AND NUMBER. 141
cause, and the rest, may not IK? termed different forms of the impulse of
the mind to generalize. If we put together all the Fundamental Ideas
as results of the Generalizing Impulse, we must still separate them as
different modes of action of that Impulse, showing themselves in various
characteristic ways in the axioms and modes of reasoning which belong
to different sciences. The Generalizing Impulse in one case proceeds
according to the Idea of Space ; in another, according to the Idea of
Mechanical Cause ; and so in other subjects.
CHAPTER XI.
OF MATHEMATICAL REASONING.
1. Discursive Reasoning. — WE have thus seen that
our notions of space, time, and their modifications, neces
sarily involve a certain activity of the mind ; and that
the conditions of this activity form the foundations of
those sciences which have the relations of space, time,
and number, for their object. Upon the fundamental
principles thus established, the various sciences which
are included in the term Pure Mathematics, (Geometry,
Algebra, Trigonometry, Conic Sections, and the rest of
the Higher Geometry, the Differential Calculus, and the
like,) are built up by a series of reasonings. These rea
sonings are subject to the rules of Logic, as we have
already remarked ; nor is it necessary here to dwell long
on the nature and rules of such processes. But we may
here notice that such processes are termed discursire,
in opposition to the operations by which we acquire our
fundamental principles, which are, as we have seen, intui
tive. This opposition was formerly very familiar to our
writers ; as Milton,—
. . . Thus the soul reason receives,
Discursive or intuitive. — Paradise Lost, \. 438.
For in such reasonings we obtain our conclusions, not
by looking at our conceptions steadily in one view, which
142 PHILOSOPHY OF THE PURE SCIENCES.
is intuition, but by passing from one view to another, like
those who run from place to place (discursus). Thus a
straight line may be at the same time a side of a triangle
and a radius of a circle : and in the first proposition of
Euclid a line is considered, first in one of these relations,
and then in the other, and thus the sides of a certain
triangle are proved to be equal. And by this " discourse
of reason," as by our older writers it was termed, we set
forth from those axioms which we perceive by intuition,
travel securely over a vast and varied region, and become
possessed of a copious store of mathematical truths.
2. Technical Terms of Reasoning. — The reasoning of
mathematics, thus proceeding from a few simple princi
ples to many truths, is conducted according to the rules
of Logic. If it be necessary, mathematical proofs may be
reduced to logical forms, and expressed in Syllogisms,
consisting of major, minor, and conclusion. But in most
cases the syllogism is of that kind which is called by
logical writers an Enthymeme; a word which implies
something existing in the thoughts only, and which desig
nates a syllogism in which one of the premises is under
stood, and not expressed. Thus we say in a mathematical
proof, " because the point c is the center of the circle AB,
AC is equal to BC ;" not stating the major, — that all lines
drawn from the center of a circle to the circumference
are equal; or introducing it only by a transient reference
to the definition of a circle. But the enthymeme is so
constantly used in all habitual forms of reasoning, that
it does not occur to us as being anything peculiar in
mathematical works.
The propositions which are proved to be generally
true are termed Theorems: but when anything is required
to be done, as to draw a line or a circle under given
conditions, this proposition is a Problem. A theorem re
quires demonstration; a problem, solution. And for both
OF MATHEMATICAL REASONING. 143
purposes the mathematician usually makes a Constrw-
tion. He directs us to draw certain lines, circles, or other
curves, on which is to be founded his demonstration that
his theorem is true, or that his problem is solved. Some
times, too, he establishes some Lemma, or preparatory
proposition, before he proceeds to his main task ; and
often he deduces from his demonstration some conclusion
in addition to that which was the professed object of his
proposition ; and this is termed a Corollary.
These technical terms are noted here, not as being
very important, but in order that they may not sound
strange and unintelligible if we should have occasion to
use some of them. There is, however, one technical dis
tinction more peculiar, and more important.
3. Geometrical Analysis and Synthesis. — In geome
trical reasoning such as we have described, we introduce
at every step some new consideration ; and it is by com
bining all these considerations, that we arrive at the
conclusion, that is, the demonstration of the proposition.
Each step tends to the final result, by exhibiting some
part of the figure under a new relation. To what we
have already proved, is added something more; and hence
this process is called Synthesis, or putting together. The
proof flows on, receiving at every turn new contribu
tions from different quarters ; like a river fed and aug
mented by many tributary streams. And each of these
tributaries flows from some definition or axiom as its
fountain, or is itself formed by the union of smaller rivulets
which have sources of this kind. In descending along its
course, the synthetical proof gathers all these accessions
into one common trunk, the proposition finally proved.
But we may proceed in a different manner. We
mav besin from the formed river, and ascend to its
•
sources. We may take the proposition of which we
require a proof, and may examine what the supposition
144 PHILOSOPHY OF THE PURE SCIENCES.
of its truth implies. If this be true, then something else
may be seen to be true ; and from this, something else,
and so on. We may often, in this way, discover of what
simpler propositions our theorem or solution is com
pounded, and may resolve these in succession, till we
come to some proposition which is obvious. This is geo
metrical Analysis. Having succeeded in this analytical
process, we may invert it ; and may descend again from
the simple and known propositions, to the proof of a
theorem, or the solution of a problem, which was our
starting-place.
This process resembles, as we have said, tracing a
river to its sources. As we ascend the stream, we per
petually meet with bifurcations ; and some sagacity is
needed to enable us to see which, in each case, is the
main stream : but if we proceed in our 'research, we
exhaust the unexplored valleys, and finally obtain a clear
knowledge of the place whence the waters flow. Analy
tical is sometimes confounded with symbolical reasoning,
on which subject we shall make a remark in the next
chapter. The object of that chapter is to notice certain
other fundamental principles and ideas, not included in
those hitherto spoken of, which we find thrown in our
way as we proceed in our mathematical speculations.
It would detain us too long, and involve us in subtle and
technical disquisitions, to examine fully the grounds of
these principles ; but the Mathematics hold so important
a place in relation to the inductive sciences, that I shall
briefly notice the leading ideas which the ulterior pro
gress of the subject involves.
145
CHAPTER XII.
OF THE FOUNDATIONS OF THE HIGHER
MATHEMATICS.
I. The Idea of a Limit. — THE general truths concern
ing relations of space which depend upon the axioms
and definitions contained in Euclid's Elements, and which
involve only properties of straight lines and circles, are
termed Elementary Geometry : all beyond this belongs to
the Higher Geometry. To this latter province appertain,
for example, all propositions respecting the lengths of any
portions of curve lines ; for these cannot be obtained by
means of the principles of the Elements alone. Here
then we mus£ ask to what other principles the geometer
has recourse, and from what source these are drawn. Is
there any origin of geometrical truth which we have not
yet explored ?
The Idea of a Limit supplies a new mode of establish
ing mathematical truths. Thus with regard to the length
of any portion of a curve, a problem which we have just
mentioned ; a curve is not made up of straight lines, and
therefore we cannot by means of any of the doctrines of
elementary geometry measure the length of any curve.
But we may make up a figure nearly resembling any
curve by putting together many short straight lines, just
as a polygonal building of very many sides may nearly
resemble a circular room. And in order to approach
nearer and nearer to the curve, we may make the sides
more and more small, more and more numerous. We
may then possibly find some mode of measurement, some
relation of these small lines to other lines, which is not
disturbed by the multiplication of the sides, however far
it be carried. And thus, we may do what is equivalent to
VOL. i. \v. P. L
14() PHILOSOPHY OF TFIE PURE SCIENCES.
measuring the curve itself; for by multiplying the sides
we may approach more and more closely to the curve till
no appreciable difference remains. The curve line is the
Limit of the polygon ; and in this process we proceed on
the Axiom, that "What is true up to the limit is true at
the limit."
This mode of conceiving mathematical magnitudes is
of wide extent and use ; for every curve may be con
sidered as the limit of some polygon ; every varied
magnitude, as the limit of some aggregate of simpler
forms ; and thus the relations of the elementary figures
enable us to advance to the properties of the most com
plex cases.
A Limit is a peculiar and fundamental conception, the
use of which in proving the propositions of the Higher
Geometry cannot be superseded by any combination of
other hypotheses and definitions*. The axiom just no
ticed, that what is true up to the limit is true at the limit,
is involved in the very conception of a limit : and this
principle, with its consequences, leads to all the results
which form the subject of the higher mathematics, whe-
* This assertion cannot be fully proved and illustrated without a
reference to mathematical reasonings which would not be generally
intelligible. I have shown the truth of the assertion in my Thoughts
on the Study of Mathematics, annexed to the Principles of English
University Education. The proof is of this kind : — The ultimate
equality of an arc of a curve and the corresponding periphery of a
polygon, when the sides of the polygon are indefinitely increased in
number, is evident. But this truth cannot be proved from any other
axiom. For if we take the supposed axiom, that a curve is always
less than the including broken line, this is not true, except with a con
dition ; and in tracing the import of this condition, we find its neces
sity becomes evident only when we introduce a reference to a Limit.
And the same is the case if we attempt to supersede the notion of a
Limit in proving any other simple and evident proposition in which
that notion is involved. Therefore these evident truths are self-evident,
in virtue of /lie Idea of a Limit.
THE FOUNDATIONS OF THE HIGHER MATHEMATICS. 147
thcr proved by the consideration of evanescent triangles,
by the processes of the Differential Calculus, or in any
other way.
The ancients did not expressly introduce this con
ception of a Limit into their mathematical reasonings ;
although in the application of what is termed the
Method of Exhaustions, (in which they show how to
exhaust the difference between a polygon and a curve, or
the like,) they were in fact proceeding upon an obscure
apprehension of principles equivalent to those of the
Method of Limits. Yet the necessary fundamental prin
ciple not having, in their time, been clearly developed,
their reasonings were both needlessly intricate and im
perfectly satisfactory. Moreover they were led to put in
the place of axioms, assumptions which were by no means
self-evident ; as when Archimedes assumed, for the basis
of his measure of the circumference of the circle, the
proposition that a circular arch is necessarily less than
two lines which inclose it, joining its extremities. The
reasonings of the older mathematicians, which professed
to proceed upon such assumptions, led to true results
in reality, only because they were guided by a latent
reference to the limiting case of such assumptions. And
this latent employment of the conception of a Limit,
reappeared in various forms during the early period of
modern mathematics ; as for example, in the Method of
Indivisibles of Cavalleri, and the Characteristic Triangle
of Barrow ; till at last, Newton distinctly referred such
reasonings to the conception of a Limit, and established
the fundamental principles and processes which that
conception introduces, with a distinctness and exactness
which required little improvement to make it as unim
peachable as the demonstrations of geometry. And when
such processes as Newton thus deduced from the con
ception of a Limit are represented by means of general
L2
148 PHILOSOPHY OF THE PURE SCIENCES.
algebraical symbols instead of geometrical diagrams, we
have then before us the Method of Fluxions, or the
Differential Calculus; a mode of treating mathematical
problems justly considered as the principal weapon by
which the splendid triumphs of modern mathematics
have been achieved.
2. The Use of General Symbols. — The employment
of algebraical symbols, of which we have just spoken,
has been another of the main instruments to which the
successes of modern mathematics are owing. And here
again the processes by which we obtain our results de
pend for their evidence upon a fundamental conception,
— the conception of arbitrary symbols as the Signs of
quantity and its relations ; and upon a corresponding
axiom, that " The interpretation of such symbols must
be perfectly general." In this case, as in the last, it was
only by degrees that mathematicians were led to a just
apprehension of the grounds of their reasoning. For
symbols were at first used only to represent numbers
considered with regard to their numerical properties ;
and thus the science of Algebra was formed. But it was
found, even in cases belonging to common algebra, that
the symbols often admitted of an interpretation which
went beyond the limits of the problem, and which yet was
not unmeaning, since it pointed out a question closely
analogous to the question proposed. This was the case,
for example, when the answer was a negative quantity ;
for when Descartes had introduced the mode of repre
senting curves by means of algebraical relations among
the symbols of the co-ordinates, or distances of each of
their points from fixed lines, it was found that negative
quantities must be dealt with as not less truly significant
than positive ones. And as the researches of mathema
ticians proceeded, other cases also were found, in which
the symbols, although destitute of meaning according to
THE FOUNDATIONS OF THE HIGHER MATHEMATICS. 140
the original conventions of their institution, still pointed
out truths which could be verified in other ways ; as in
the cases in which what are called Impossible quantities
occur. Such processes may usually be confirmed upon
other principles, and the truth in question may be esta
blished by means of a demonstration in which no such
secerning fallacies defeat the reasoning. But it has also
been shown in many such cases, that the process in which
some of the steps appear to be without real meaning,
does in fact involve a valid proof of the proposition.
And what we have here to remark is, that this is not
true accidentally or partially only, but that the results
of systematic symbolical reasoning must always express
general truths, by their nature, and do not, for their
justification, require each of the steps of the process to
represent some definite operation upon quantity. The
absolute universality of the interpretation of symbols is
the fundamental principle of their use. This has been
shown very ably by Dr. Peacock in his Alyebra. He
has there illustrated, in a variety of ways, this prin
ciple : that " If general symbols express an identity
when they are supposed to be of any special nature,
they must also express an identity when they are gene
ral in their nature." And thus, this universality of sym
bols is a principle in addition to those we have already
noticed ; and is a principle of the greatest importance
in the formation of mathematical science, according to
the wide generality which such science has in modern
times assumed.
3. Connexion of Symbols and Analysis. — Since in
our symbolical reasoning our symbols thus reason for us,
we do not necessarily here, as in geometrical reasoning,
go on adding carefully one known truth to another, till
we reach the desired result. On the contrary, if we have
a theorem to prove or a problem to solve which can be
150 PHILOSOPHY OF THE PUKE SCIENCES.
brought under the domain of our symbols, we may at
once state the given but unproved truth, or the given
combination of unknown quantities, in its symbolical
form. After this first process, we may then proceed to
trace, by means of our symbols, what other truth is
involved in the one thus stated, or what the unknown
symbols must signify; resolving step by step the sym
bolical assertion with which we began, into others more
fitted for our purpose. The former process is a kind of
synthesis, the latter is termed analysis. And although
symbolical reasoning does not necessarily imply such
analysis; yet the connexion is so familiar, that the
term analysis is frequently used to designate symbolical
reasoning.
CHAPTER XIII.
THE DOCTRINE OF MOTION.
1. Pure Mechanism,. — THE doctrine of Motion, of
which we have here to speak, is that in which motion is
considered quite independently of its cause, force; for
all consideration of force belongs to a class of ideas
entirely different from those with which we are here
concerned. In this view it may be termed the pure
doctrine of motion, since it has to do solely with space
and time, which are the subjects of pure mathematics.
(See C. I. of this Book.) Although the doctrine of
motion in connexion with force, which is the subject
of mechanics, is by far the most important form in
which the consideration of motion enters into the form
ation of our sciences, the Pure Doctrine of Motion,
which treats of space, time, and velocity, might be fol
lowed out so as to give rise to a very considerable and
curious body of science. Such a science is the science
THE DOCTRINE OF MOTION. 151
of Mechanism, independent of force, and considered as
the solution of a problem which may be thus enunciated:
*' To communicate any given motion from a first mover
to a given body." The science which should have for its
object to solve all the various cases into which this pro
blem would ramify, might be termed Pure Mechanism,
in contradistinction to Mechanics Proper, or Machinery,
in which Force is taken into consideration. The greater
part of the machines which have been constructed for
use in manufactures have been practical solutions of some
of the cases of this problem. We have also important
contributions to such a science in the works of mathe
maticians ; for example, the various investigations and
demonstrations which have been published respecting
the form of the Teeth of Wheels, and Mr. Babbage's
memoir* on the Language of Machinery. There are
also several works which contain collections of the
mechanical contrivances which have been invented for
the purpose of transmitting and modifying motion, and
these works may be considered as treatises on the science
of Pure Mechanism. But this science has not yet been
reduced to the systematic simplicity wfhich is desirable,
nor indeed generally recognized as a separate science. It
has been confounded, under the common name of Me
chanics, with the other science, Mechanics Proper, or
Machinery, which considers the effect of force transmitted
by mechanism from one part of a material combination
to another. For example, the Mechanical Powers, as
they are usually termed, (the Lever, the Wheel and
Axle, the Inclined Plane, the Wedge, and the Screw,)
have almost always been treated with reference to the
relation between the Power and the Weight, and not
primarily as a mode of changing the velocity and kind
* On a Method of expressing by Signs l/ic Action of Machinery.
Phil. Trans., 1826, p. 250.
PHILOSOPHY OF THE PURE SCIENCES.
of the motion. The science of pure motion has not
generally been separated from the science of motion
viewed with reference to its causes.
Recently, indeed, the necessity of such a separation
has been seen by those who have taken a philosophical
view of science. Thus this necessity has been urged by
M. Ampere, in his Essai sur la Philosophic, des Sciences
(1834): "Long," he says, (p. 50), "before I employed
myself upon the present work, I had remarked that it is
usual to omit, in the brginning of all books treating of
sciences which regard motion and force, certain consi
derations which, duly developed, must constitute a special
science : of which science certain parts have been treated
of, either in memoirs or in special works ; such, for ex
ample, as that of Carnot upon Motion considered geome
trically, and the essay of Lanz and Betancourt upon the
Composition of Machines." He then proceeds to describe
this science nearly as we have done, and proposes to
term it Kinematics (Cinematique], from /ai^a, motion.
2. Formal Astronomy. — I shall not attempt here
further to develop the form which such a science must
assume. But I may notice one very large province which
belongs to it. When men had ascertained the apparent
motions of the sun, moon, and stars, to a moderate
degree of regularity and accuracy, they tried to conceive
in their minds some mechanism by which these motions
might be produced ; and thus they in fact proposed to
themselves a very extensive problem in Kinematics.
This, indeed, was the view originally entertained of the
nature of the science of astronomy. Thus Plato in the
seventh Book of his Republic*, speaks of astronomy as
the doctrine of the motion of solids, meaning thereby,
spheres. And the same was a proper description of the
science till the time of Kepler, and even later: for
4 P. 528.
THE DOCTRINE OF MOTION. 153
Kepler endeavoured in vain to conjoin with the know
ledge of the motions of the heavenly bodies, those true
mechanical conceptions which converted formal into
physical astronomy"".
The astronomy of the ancients admitted none but
uniform circular motions, and could therefore be com
pletely cultivated by the aid of their elementary geo
metry. But the pure science of motion might be
extended to all motions, however varied as to the speed
or the path of the moving body. In this form it must
depend upon the doctrine of limits ; and the funda
mental principle of its reasonings would be this : That
velocity is measured by the Limit of the space described,
considered with reference to the time in which it is
described. I shall not further pursue this subject ; and
in order to complete what I have to say respecting the
Pure Sciences, I have only a few words to add respect
ing their bearing on Inductive Science in general.
CHAPTER XIV.
OF THE APPLICATION OF MATHEMATICS TO
THE INDUCTIVE SCENCES.
I. ALL objects in the world which can be made the
subjects of our contemplation are subordinate to the
conditions of Space, Time, and Number; and on this
account, the doctrines of pure mathematics have most
numerous and extensive applications in every depart
ment of our investigations of nature. And there is a
peculiarity in these Ideas, which has caused the mathe
matical sciences to be, in all cases, the first successful
efforts of the awakening speculative powers of nations at
* Hist. Ind. ,SY.. ii. 130.
154 PHILOSOPHY OF THE PURE SCIENCES.
the commencement of their intellectual progress. Con
ceptions derived from these Ideas are, from the very
first, perfectly precise and clear, so as to be fit elements
of scientific truths. This is not the case with the other
conceptions which form the subjects of scientific in
quiries. The conception of statical force, for instance,
was never presented in a distinct form till the works of
Archimedes appeared : the conception of accelerating
force was confused, in the mind of Kepler and his con
temporaries, and only became clear enough for purposes
of sound scientific reasoning in the succeeding century :
the just conception of chemical composition of elements
gradually, in modern times, emerged from the erroneous
and vague notions of the ancients. If we take works
published on such subjects before the epoch when the
foundations of the true science were laid, we find the
knowledge not only small, but worthless. The writers
did not see any evidence in what we now consider as the
axioms of the science ; nor any inconsistency where we
now see self-contradiction. But this was never the case
with speculations concerning space and number. From
their first rise, these were true as far as they went.
The Geometry and Arithmetic of the Greeks and Indians,
even in their first and most scanty form, contained none
but true propositions. Men's intuitions upon these sub
jects never allowed them to slide into error and confu
sion ; and the truths to which they were led by the first
efforts of their faculties, so employed, form part of the
present stock of our mathematical knowledge.
2, But we are here not so much concerned with
mathematics in their pure form, as with their applica
tion to the phenomena and laws of nature. And here
also the very earliest history of civilization presents to
us some of the most remarkable examples of man's suc
cess in his attempts to attain to science. Space and
INDUCTIVE APPLICATION OF MATHEMATICS. 155
time, position and motion, govern all visible objects ;
but by far the most conspicuous examples of the rela
tions which arise out of such elements, are displayed by
the ever-moving luminaries of the sky, which measure
days, and months, and years, by their motions, and
man's place on the earth by their position. Hence the
sciences of space and number were from the first culti
vated with peculiar reference to Astronomy. I have
elsewhere * quoted Plato's remark, — that it is absurd
to call the science of the relations of space geometry,
the measure of the earth, since its most important office
is to be found in its application to the heavens. And
on other occasions also it appears how strongly he, who
may be considered as the representative of the scientific
and speculative tendencies of his time and country, had
been impressed with the conviction, that the formation
of a science of the celestial motions must depend entirely
upon the progress of mathematics. In the Epilogue to
the Dialogue on the Laws\, he declares mathematical
knowledge to be the first and main requisite for the
astronomer, and describes the portions of it which he
holds necessary for astronomical speculators to culti
vate. These seem to be, Plane Geometry, Theoretical
Arithmetic, the Application of Arithmetic to planes
and to solids, and finally the doctrine of Harmonics.
Indeed the bias of Plato appears to be rather to con
sider mathematics as the essence of the science of
astronomy, than as its instrument ; and he seems dis
posed, in this as in other things, to disparage observa
tion, and to aspire after a science founded upon demon
stration alone. " An astronomer," he says in the same
place, "must not be like Hesiod and persons of that
kind, whose astronomy consists in noting the settings
and risings of the stars ; but he must be one who
* ///*/. Ind. Sc.. D. in. c. ii. t Epinomis. p. 990.
156 PHILOSOPHY OF THE PURE SCIENCES.
understands the revolutions of the celestial spheres, each
performing its proper cycle."
A large portion of the mathematics of the Greeks,
so long as their scientific activity continued, was directed
towards astronomy. Besides many curious propositions
of plane and solid Geometry, to which their astronomers
were led, their Arithmetic, though very inconvenient in
its fundamental assumptions, was cultivated to a great
extent ; and the science of Trigonometry, in which pro
blems concerning the relations of space were resolved by
means of tables of numerical results previously obtained,
was created. Menelaus of Alexandria wrote six Books
on Chords, probably containing methods of calculating
Tables of these quantities ; such Tables were familiarly
used by the later Greek astronomers. The same author
also wrote three Books on Spherical Trigonometry,
which are still extant.
3. The Greeks, however, in the first vigour of their
pursuit of mathematical truth, at the time of Plato and
soon after, had by no means confined themselves to
those propositions which had a visible bearing on the
phenomena of nature ; but had followed out many beau
tiful trains of research, concerning various kinds of
figures, for the sake of their beauty alone ; as for in
stance in their doctrine of Conic Sections, of which
curves they had discovered all the principal properties.
But it is curious to remark, that these investigations,
thus pursued at first as mere matters of curiosity and
intellectual gratification, were destined, two thousand
years later, to play a very important part in establishing
that system of the celestial motions which succeeded the
Platonic scheme of cycles and epicycles. If the proper
ties of the conic sections had not been demonstrated by
the Greeks, and thus rendered familiar to the mathe
maticians of succeeding ages, Kepler would probably
APPLICATION OF MATIIKMATICS. 157
not have been able to discover those laws respecting the
orbits and motions of the planets which were the occa
sion of the greatest revolution that ever happened in
the history of science.
4. The Arabians, who, as I have elsewhere said,
added little of their own to the stores of science which
they received from the Greeks, did however make some
very important contributions in those portions of pure
mathematics which are subservient to astronomy. Their
adoption of the Indian mode of computation by means
of the Ten Digits, 1, 2, 3, 4, 5, G, 7, 8, 9, 0, and by the
method of Local Values, instead of the cumbrous sexa
gesimal arithmetic of the Greeks, was an improvement
by which the convenience and facility of numerical cal
culations were immeasurably augmented. The Arabians
also rendered several of the processes of trigonometry
much more commodious, by using the Sine of an arc
instead of the Chord ; an improvement which Albateg-
nius appears to claim for himself'-'; and by employing
also the Tangents of arcs, or, as they called themf,
upright shadows.
5. The constant application of mathematical know
ledge to the researches of Astronomy, and the mutual
influence of each science on the progress of the other,
has been still more conspicuous in modern times. New
ton's Method of Prime and Ultimate Ratios, which we
have already noticed as the first correct exposition of
the doctrine of a Limit, is stated in a series of Lemmas,
or preparatory theorems, prefixed to his Treatise on the
System of the World. Both the properties of curve
lines and the doctrines concerning force and motion,
which he had to establish, required that the common
mathematical methods should be methodized and ex
tended. If Newton had not been a most expert and in-
* Delambre, Ast, M. A., p. 12. + I hid., p. 17.
158 PHILOSOPHY OF THE PURE SCIENCES.
ventive mathematician, as well as a profound and philo
sophical thinker, he could never have made any one of
those vast strides in discovery of which the rapid succes
sion in his work strikes us with wonder*. And if we
see that the great task begun by him, goes on more
slowly in the hands of his immediate successors, and
lingers a little before its full completion, we perceive
that this arises, in a great measure, from the defect of
the mathematical methods then used. Newton's syn
thetical modes of investigation, as we have elsewhere
observed, were an instrument f, powerful indeed in his
mighty hand, but too ponderous for other persons to
employ with effect. The countrymen of Newton clung
to it the longest, out of veneration for their master ; and
English cultivators of physical astronomy were, on that
very account, left behind the progress of mathematical
science in France and Germany, by a wide interval,
which they have only recently recovered. On the Conti
nent, the advantages offered by a familiar use of symbols,
and by attention to their symmetry and other relations,
were accepted without reserve. In this manner the
Differential Calculus of Leibnitz, which was in its origin
and signification identical with the Method of Fluxions
of Newton, soon surpassed its rival in the extent and
generality of its application to problems. This Calculus
was applied to the science of mechanics, to which it,
along with the symmetrical use of co-ordinates, gave a
new form ; for it was soon seen that the most difficult
problems might in general be reduced to finding inte
grals, which is the reciprocal process of that by which
differentials are found ; so that all difficulties of physical
astronomy were reduced to difficulties of symbolical cal
culation, these, indeed, being often sufficiently stubborn.
Clairaut, Euler, and D'Alembert employed the increased
* Hist. Ind. $c., R. vn. c. ii. t //,., p. 175.
INDUCTIVE APPLICATION OF MATHEMATICS. 159
resources of mathematical science upon the Theory of
the Moon, and other questions relative to the system of
the world ; and thus began to pursue such inquiries in
the course in which mathematicians are still labouring
up to the present day. This course was not without its
checks and perplexities. We have elsewhere quoted*
Clairaut's expression when he had obtained the very
complex differential equations which contain the solu
tion of the problem of the moon's motion :' " Now inte
grate them who can !" But in no very long time they
were integrated, at least approximately ; and the methods
of approximation have since then been improved ; so
that now, with a due expenditure of labour, they may be
carried to any extent which is thought desirable. If
the methods of astronomical observation should here
after reach a higher degree of exactness than they now
profess, so that irregularities in the motions of the sun,
moon, and planets, shall be detected which at present
escape us, the mathematical part of the theory of univer
sal gravitation is in such a condition that it can soon be
brought into comparison with the newly-observed facts.
Indeed at present the mathematical theory is in advance
of such observations. It can venture to suggest what
may afterwards be detected, as well as to explain what
has already been observed. This has happened recently;
for Professor Airy has calculated the law and amount
of an inequality depending upon the mutual attraction of
the Earth and Venus ; of which inequality (so small is
it,) it remains to be determined whether its effect can be
traced in the series of astronomical observations.
G. As the influence of mathematics upon the progress
of astronomy is thus seen in the cases in which theory
and observation confirm each other, so this influence ap
pears in another way, in the very few cases in which the
* Hist. Ind. Sc., R vi. c. vi. sect. 7-
1()0 PHILOSOPHY OF THE PUUK SCIENCES.
facts have not been fully reduced to an agreement with
theory. The most conspicuous case of this kind is the
state of our knowledge of the Tides. This is a portion
of astronomy : for the Newtonian theory asserts these
curious phenomena to be the result of the attraction of
the sun and moon. Nor can there be any doubt that
this is true, as a general statement ; yet the subject is
up to the present time a blot on the perfection of the
theory of universal gravitation ; for we are very far from
being able in this, as in the other parts of astronomy, to
show that theory will exactly account for the time, and
magnitude, and all other circumstances of the pheno
menon at every place on the earth's surface. And what
is the portion of our mathematics which is connected
with this solitary signal defect in astronomy ? It is the
mathematics of the Motion of Fluids ; a portion in which
extremely little progress has been made, and in which all
the more general problems of the subject have hitherto
remained entirely insoluble. The attempts of the greatest
mathematicians, Newton, Maclaurin, Bernoulli, Clairaut,
Laplace, to master such questions, all involve some gra
tuitous assumption, which is introduced because the
problem cannot otherwise be mathematically dealt with :
these assumptions confessedly render the result defective,
and how defective, it is hard to say. And it was pro
bably precisely the absence of a theory which could be
reasonably expected to agree with the observations, which
made Observations of this very curious phenomenon, the
Tides, to be so much neglected as till very recently they
were. Of late years such observations have been pur
sued, and their results have been resolved into empirical
laws, so that the rules of the phenomena have been
ascertained, although the dependence of these rules upon
the lunar and solar forces has not been shown. Here
then we have a portion of our knowledge relating to
INDUCTIVE APPLICATION OF MATHEMATICS. 1(11
facts undoubtedly dependent upon universal gravitation,
in which Observation has outstripped Theory in her pro
gress, and is compelled to wait till her usual companion
overtakes her. This is a position of which Mathematical
Theory has usually been very impatient, and we may
expect that she will be no less so in the present instance.
7. It would be easy to show from the history of
other sciences, for example, Mechanics and Optics, how
essential the cultivation of pure mathematics has been to
their progress. The parabola was already familiar among
mathematicians when Galileo discovered that it was the
theoretical path of a Projectile ; and the extension and
generalization of the Laws of Motion could never have
been effected, unless the Differential and Integral Cal
culus had been at hand, ready to trace the results of every
hypothesis which could be made. D'Alembert's mode of
expressing the Third Law of Motion in its most general
form"''", if it did not prove the law, at least reduced the
application of it to analytical processes which could be
performed in most of those cases in which they were
needed. In many instances the demands of mechanical
science suggested the extension of the methods of pure
analysis. The problem of Vibrating Strings gave rise to
the Calculus of Partial Differences, which was still fur
ther stimulated by its application to the motions of fluids
and other mechanical problems. And we have in the
writings of Lagrange and Laplace other instances equally
remarkable of new analytical methods, to \vhich mecha
nical problems, and especially cosmical problems, have
given occasion.
8. The progress of Optics as a science has, in like
manner, been throughout dependent upon the progress
<>i' pure mathematics. The first rise of geometry was fol-
* Hi.il. hid. Sd., R vi. c. vi. sort. 7-
VOL. i. w. r. M
1()2 PHILOSOPHY OI-: Till-: PUUK SCIKNCKS.
lowed by some advances, slight ones no doubt, in the
doctrine of Reflection and in Perspective. The law of
Refraction was traced to its consequences by means of
Trigonometry, which indeed was requisite to express the
law in a simple form. The steps made in Optical science
by Descartes, Newton, Euler, and Huyghens, required
the geometrical skill which those philosophers possessed.
And if Young and Fresnel had not been, each in his
peculiar way, persons of eminent mathematical endow
ments, they would not have been able to bring the
Theory of Undulations and Interferences into a condi
tion in which it could be tested by experiments. We
may see how unexpectedly recondite parts of pure mathe
matics may bear upon physical science, by calling to
mind a circumstance already noticed in the History of
Science* ; — that Fresnel obtained one of the most curious
confirmations of the theory (the laws of Circular Polar
ization by reflection) through an interpretation of an
algebraical expression, which, according to the original
conventional meaning of the symbols, involved an im
possible quantity. We have already remarked, that in
virtue of the principle of the generality of symbolical
language, such an interpretation may often point out
some real and important analogy.
9. From this rapid sketch it may be seen how
important an office in promoting the progress of the
physical sciences belongs to mathematics. Indeed in
the progress of many sciences, every step has been so
intimately connected with some advance in mathematics,
that we can hardly be surprized if some persons have
considered mathematical reasoning to be the most essen
tial part of such sciences ; and have overlooked the other
elements which enter into their formation. How erro-
* Hist. Ind. <Scz'., B. ix. c. xiii. sect. 2.
INDUCTIVE APPLICATION OF MATHEMATICS. 163
neous this view is we shall best see by turning our
attention to the other Ideas besides those of space, num
ber, and motion, which enter into some of the most
conspicuous and admired portions of what is termed
exact science ; and by showing that the clear and distinct
developement of such Ideas is quite as necessary to the
progress of exact and real knowledge as an acquaintance
with arithmetic and geometry.
M a
164
BOOK III.
THE PHILOSOPHY OF THE MECHANICAL
SCIENCES.
CHAPTER I.
OF THE MECHANICAL SCIENCES.
IN the History of the Sciences, that class of which we
here speak occupies a conspicuous and important place ;
coming into notice immediately after those parts of astro
nomy which require for their cultivation merely the
ideas of space, time, motion, and number. It appears
from our History, that certain truths concerning the equi
librium of bodies were established by Archimedes ; — that,
after a long interval of inactivity, his principles were
extended and pursued further in modern times : — and
that to these doctrines concerning equilibrium and the
forces which produce it, (which constitute the science
Statics,} were added many other doctrines concerning
the motions of bodies, considered also as produced by
forces, and thus the science of Dynamics was produced.
The assemblage of these sciences composes the province
of Mechanics. Moreover, philosophers have laboured to
make out the laws of the equilibrium of fluid as well as
solid bodies ; and hence has arisen the science of Hydro
statics. And the doctrines of Mechanics have been found
to have a most remarkable bearing upon the motions
of the heavenly bodies ; with reference to which, indeed,
they were at first principally studied. The explanation
OF THE MECHANICAL SCIENCES. 105
of those cosmical facts by means of mechanical principles
and their consequences, forms the science of Physical
Astronomy. These are the principal examples of mecha
nical science ; although some other portions of Physics,
as Magnetism and Electrodynamics, introduce mecha
nical doctrines very largely into their speculations.
Now in all these sciences we have to consider Forces.
In all mechanical reasonings forces enter, either as pro
ducing motion, or as prevented from doing so by other
forces. Thus force, in its most general sense, is the cause
of motion, or of tendency to motion ; and in order to
discover the principles on which the mechanical sciences
truly rest, we must examine the nature and origin of
our knowledge of Causes.
In these sciences, however, we have not to deal with
Cause in its more general acceptation, in which it applies
to all kinds of agency, material or immaterial ; — to the
influence of thought and will, as well as of bodily pres
sure and attractive force. Our business at present is
only with such causes as immediately operate upon
matter. We shall nevertheless, in the first place, con
sider the nature of Cause in its most general form ; and
afterwards narrow our speculations so as to direct them
specially to the mechanical sciences.
CHAPTER II.
OF THE IDEA OF CAUSE.
1. WE see in the world around us a constant suc
cession of causes and effects connected with each other.
The laws of this connexion we learn in a great measure
from experience, by observation of the occurrences which
present themselves to our notice, succeeding one another.
166 PHILOSOPHY OF THE MECHANICAL SCIENCES.
But in doing this, and in attending to this succession of
appearances, of which we are aware by means of our
senses, we supply from our own minds the Idea of Cause.
This Idea, as we have already shown with respect to
other Ideas, is not derived from experience, but has its
origin in the mind itself; — is introduced into our expe
rience by the active, and not by the passive part of our
nature.
By Cause we mean some quality, power, or efficacy,
by which a state of things produces a succeeding state.
Thus the motion of bodies from rest is produced by a
cause which we call Force : and in the particular case
in which bodies fall to the earth, this force is termed
Gravity. In these cases, the Conceptions of Force and
Gravity receive their meaning from the Idea of Cause
which they involve : for Force is conceived as the Cause
of Motion. That this Idea of Cause is not derived from
experience, we prove (as in former cases) by this con
sideration : that we can make assertions, involving this
idea, which are rigorously necessary and universal ;
whereas knowledge derived from experience can only be
true as far as experience goes, and can never contain in
itself any evidence whatever of its necessity. We assert
that " Every event must have a cause :" and this proposi
tion we know to be true, not only probably, and gene
rally, and as far as we can see : but we cannot suppose
it to be false in any single instance. We are as certain
of it as of the truths of arithmetic or geometry. We
cannot doubt that it must apply to all events past and
future, in every part of the universe, just as truly as
to those occurrences which we have ourselves observed.
What causes produce what effects; — what is the cause
of any particular event ; — what will be the effect of any
peculiar process ; — these are points on which experience
may enlighten us. Observation and experience may be
OF THE IDEA OF CAUSE. 107
requisite, to enable us to judge respecting such matters.
But that every event has some cause, Experience cannot
prove any more than she can disprove. She can add
nothing to the evidence of the truth, however often she
may exemplify it. This doctrine, then, cannot have been
acquired by her teaching ; and the Idea of Cause, which
the doctrine involves, and on which it depends, cannot
have come into our minds from the region of observa
tion.
2. That we do, in fact, apply the Idea of Cause in a
more extensive manner than could be justified, if it were
derived from experience only, is easily shown. For from
the principle that everything must have a cause, we not
only reason concerning the succession of the events which
occur in the progress of the world, and which form the
course of experience ; but we infer that the world itself
must have a cause ; — that the chain of events connected
by common causation, must have a First Cause of a
nature different from the events themselves. This we
are entitled to do, if our Idea of Cause be independent of,
and superior to, experience : but if we have no Idea of
Cause except such as we gather from experience, this
reasoning is altogether baseless and unmeaning.
3. Again ; by the use of our powers of observation,
we are aware of a succession of appearances and events.
But none of our senses or powers of external observa
tion can detect in these appearances the power or quality
which we call Cause. Cause is that which connects one
event with another ; but no sense or perception discloses
to us, or can disclose, any connexion among the events
which we observe. We see that one occurrence follows
another, but we can never see anything which shows that
one occurrence must follow another. We have already
noticed*, that this truth has been urged by nictaphv-
Book I., chap. xiii.
168 PHILOSOPHY OF THE MECHANICAL SCIENCES.
sicians in modern times, and generally assented to by
those who examine carefully the connexion of their own
thoughts. The arguments are, indeed, obvious enough.
One ball strikes another and causes it to move forwards.
But by what compulsion? Where is the necessity? If
the mind can see any circumstance in this case which
makes the result inevitable, let this circumstance be
pointed out. But, in fact, there is no such discoverable
necessity ; for we can conceive this event not to take
place at all. The struck ball may stand still, for aught
we can see. " But the laws of motion will not allow it
to do so." Doubtless they will not. But the laws of
motion are learnt from experience, and therefore can
prove no necessity. Why should not the laws of motion
be other than they are? Are they necessarily true?
That they are necessarily such as do actually regulate the
impact of bodies, is at least no obvious truth ; and there
fore this necessity cannot be, in common minds, the
ground of connecting the impact of one ball with the
motion of another. And assuredly, if this fail, no other
ground of such necessary connexion can be shown. In
this case, then, the events are not seen to be necessarily
connected. But if this case, where one ball moves another
by impulse, be not an instance of events exhibiting a
necessary connexion, we shall look in vain for any ex
ample of such a connexion. There is, then, no case in
which events can be observed to be necessarily con
nected : our idea of causation, which implies that the
event is necessarily connected with the cause, cannot be
derived from observation.
4. But it may be said, we have not any such Idea of
Cause, implying necessary connexion with effect, and a
quality by which this connexion is produced. We see
nothing but the succession of events ; and by muse we
mean nothing but a certain succession of events; — name-
OF THE IDEA OF CAUSE. 1 GO
ly, a constant, unvarying succession. Cause and effect
are only two events of which the second invariably
follows the first. We delude ourselves when we ima
gine that our idea of causation involves anything more
than this.
To this I reply by asking, what then is the meaning
of the maxim above quoted, and allowed by all to be
universally and necessarily true, that every event must
have a cause ? Let us put this maxim into the language
of the explanation just noticed ; and it becomes this :—
"Every event must have a certain other event invariably
preceding it." But why must it? Where is the neces
sity ? Why must like events always be preceded by like,
except so far as other events interfere ? That there is
such a necessity, no one can doubt. All will allow that
if a stone ascend because it is thrown upwards in one
case, a stone which ascends in another case has also
been thrown upwards, or has undergone some equi
valent operation. All will allow that in this sense,
every kind of event must have some other specific kind
of event preceding it. But this turn of men's thoughts
shows that they see in events a connexion which is not
mere succession. They see in cause and effect, not
merely what does, often or always, precede and follow,
but what must precede and follow. The events are not
only conjoined, they are connected. The cause is more
than the prelude, the effect is more than the sequel, of
the fact. The cause is conceived not as a mere occa
sion ; it is a power, an efficacy, which has a real ope
ration.
5. Thus we have drawn from the maxim, that Every
Kffect must have a Cause, arguments to show that we
have an Idea of Cause which is not borrowed from expe
rience, and which involves more than mere succession.
Similar arguments might be derived from any other
170 PHILOSOPHY OF THE MECHANICAL SCIENCES.
maxims of universal and necessary validity, which we
can obtain concerning Cause : as, for example, the max
ims that Causes are measured by their Effects, and that
Reaction is equal and opposite to Action. These maxims
we shall soon have to examine ; but we may observe here,
that the necessary truth which belongs to them, shows
that they, and the Ideas which they involve, are not the
mere fruits of observation; while their meaning, including,
as it does, something quite different from the mere con
ception of succession of events, proves that such a con
ception is far from containing the whole import and
signification of our Idea of Cause.
The progress of the opinions of philosophers on the
points discussed in this chapter, has been one of the
most remarkable parts of the history of Metaphysics in
modern times : and I shall therefore briefly notice some
of its features.
CHAPTER III.
MODERN OPINIONS RESPECTING THE IDEA
OF CAUSE.
1. TOWARDS the end of the seventeenth century there
existed in the minds of many of the most vigorous and
active speculators of the European literary world, a strong
tendency to ascribe the whole of our Knowledge to the
teaching of Experience. This tendency, with its conse
quences, including among them the reaction which was
produced when the tenet had been pushed to a length
manifestly absurd, has exercised a very powerful in
fluence upon the progress of metaphysical doctrines up
to the present time. I proceed to notice some of the
most prominent of the opinions which have thus ob-
OPINIONS RESPECTING THE IDEA OF CAUSE. 171
tained prevalence among philosophers, so far as the Idea
of Cause is concerned.
Locke was one of the metaphysicians who produced
the greatest effect in diffusing this opinion, of the exclu
sive dependence of our knowledge upon experience.
Agreeably to this general system, he taught * that our
ideas of Cause and Effect are got from observation of
the things about us. Yet notwithstanding this tenet of
his, he endeavoured still to employ these ideas in rea
soning on subjects which are far beyond all limits of
experience : for he professed to prove, from our idea of
Causation, the existence of the Deity f.
Hume noticed this obvious inconsistency; but declared
himself unable to discover any remedy for a defect so
fatal to the most important parts of our knowledge. He
could see, in our belief of the succession of cause and
effect, nothing but the habit of associating in our minds
what had often been associated in our experience. He
therefore maintained that we could not, with logical
propriety, extend our belief of such a succession to cases
entirely distinct from all those of which our experience
consisted. We see, he said, an actual conjunction of two
events ; but we can in no way detect a necessary con
nexion ; and therefore we have no means of inferring
cause from effect, or effect from causej. The only way
in which we recognize Cause and Effect in the field of
our experience, is as an unfailing Sequence : we look in
vain for anything which can assure us of an infallible
Consequence. And since experience is the only source
of our knowledge, we cannot with any justice assert
that the world in which we live must necessarily have
had a cause.
2. This doctrine, taken in conjunction with the known
* Essay on the Human Understanding, B. it. c xxvi. t B. iv. c. x.
+ Hume's Phil, of the Human Mind, Vol. I. p. 94.
172 PHILOSOPHY OF Til 1C MECHANICAL SCIENCES.
skepticism of its author on religious points, produced a
considerable fermentation in the speculative world. The
solution of the difficulty thus thrown before philosophers,
was by no means obvious. It was vain to endeavour to
find in experience any other property of a Cause, than a
constant sequence of the effect. Yet it was equally vain
to try to persuade men that they had no idea of Cause ;
or even to shake their belief in the cogency of the fami
liar arguments concerning the necessity of an original
cause of all that is and happens. Accordingly these
hostile and apparently irreconcilable doctrines, — the in
dispensable necessity of a cause of every event, and the
impossibility of our knowing such a necessity, — were at
last allowed to encamp side by side. Reid, Beattie, and
others, formed one party, who showed how widely and
constantly the idea of a cause pervades all the processes
of the human mind : while another sect, including Brown,
and apparently Stewart, maintained that this idea is
always capable of being resolved into a constant se
quence ; and these latter reasoners tried to obviate the
dangerous and shocking inferences which some persons
might try to draw from their opinion, by declaring the
maxim that "Every event must have a cause," to be an
instinctive law of belief, or a fundamental principle of
the human mind*.
3. While this series of discussions was going on in
Britain, a great metaphysical genius in Germany was
unravelling the perplexity in another way. Kant's spe
culations originated, as he informs us, in the trains of
thought to which Hume's writings gave rise ; and the
Kritik der Reinen Vernunft, or Examination of the
Pure Reason, was published in 1787, with the view of
showing the true nature of our knowledge.
* Stewart's Active Powers, Vol. i. p. 347- Brown's Lectures,
Vol. i. p. 115.
OPINIONS RESPECTING THE IDEA OF CAUSE. 1 7\]
Kant's solution of the difficulties just mentioned
differs materially from that above stated. According to
Brown"-, succession observed and cause inferred, — the
memory of past conjunctions of events and the belief of
similar future conjunctions, — are facts, independent, so
far as we can discover, but inseparably combined by a
law of our mental nature. According to Kant, causality
is an inseparable condition of our experience : a con
nexion in events is requisite to our apprehending them as
events. Future occurrences must be connected by causa
tion as the past have been, because we cannot think of
past, present, and future, without such connexion. We
cannot fix the mind upon occurrences, without including
these occurrences in a series of causes and effects. The
relation of Causation is a condition under which we
think of events, as the relations of space are a condition
under which we see objects.
4. On a subject so abstruse, it is not easy to make
our distinctions very clear. Some of Brown's illustrations
appear to approach very near to the doctrine of Kant.
Thus he saysf, "The form of bodies is the relation of
their elements to each other in space, — the power of
bodies is their relation to each other in time." Yet not
withstanding such approximations in expression, the
Kantian doctrine appears to be different from the views
of Stewart and Brown, as commonly understood. Ac
cording to the Scotch philosophers, the cause and the
effect are two things, connected in our minds by a law
of our nature. But this view requires us to suppose that
we can conceive the law to be absent, and the course of
events to be unconnected. If we can understand what is
the special force of this law, we must be able to imagine
what the case would be if the law were non-existing. We
must be able to conceive a mind which does not connect
* Led.. Vol. i. ]>. 114. t Led., i. ]>. 127.
174 PHILOSOPHY OF THE MECHANICAL SCIENCES.
effects with causes. The Kantian doctrine, on the other
hand, teaches that we cannot imagine events liberated
from the connexion of cause and effect : this connexion is
a condition of our conceiving any real occurrences : we
cannot think of a real sequence of things, except as in
volving the operation of causes. In the Scotch system,
the past and the future are in their nature independent,
but bound together by a rule ; in the German system,
they share in a common nature and mutual relation, by
the act of thought which makes them past and future.
In the former doctrine cause is a tie which binds ; in the
latter it is a character which pervades and shapes events.
The Scotch metaphysicians only assert the universality
of the relation ; the German attempts further to explain
its necessity.
This being the state of the case, such illustrations as
that of Dr. Brown quoted above, in which he represents
cause as a relation of the same kind with form, do not
appear exactly to fit his opinions. Can the relations of
figure be properly said to be connected with each other
by a law of our nature, or a tendency of our mental con
stitution? Can we ascribe it to a law of our thoughts,
that we believe the three angles of a triangle to be equal
to two right angles? If so, we must give the same
reason for our belief that two straight lines cannot
inclose a space ; or that three and two are five. But
will any one refer us to an ultimate law of our consti
tution for the belief that three and two are five ? Do
we not see that they are so, as plainly as we see that
they are three and two ? Can we imagine laws of our
constitution abolished, so that three and two shall make
something different from five ; — so that an inclosed space
shall lie between two straight lines ; — so that the three
angles of a plane triangle shall be greater than two
right angles? We cannot conceive this. If the mini-
OPINIONS RESPECTING THE IDEA OF CAUSE. 1 7 ">
bers are three and two ; if the lines are straight ; if the
triangle is a rectilinear triangle, the consequences are
inevitable. We cannot even imagine the contrary. We
do not want a law to direct that things should be what
they are. The relation, then, of cause and effect, being
of the same kind as the necessary relations of figure and
number, is not properly spoken of as established in our
minds by a special law of our constitution : for we reject
that loose and inappropriate phraseology which speaks
of the relations of figure and number as " determined by
laws of belief."
5. In the present work, we accept and adopt, as the
basis of our inquiry concerning our knowledge, the exist
ence of necessary truths concerning causes, as there exist
necessary truths concerning figure and number. We
find such truths universally established and assented to
among the cultivators of science, and among speculative
men in general. All mechanicians agree that reaction
is equal and opposite to action, both when one body
presses another, and when one body communicates mo
tion to another. All reasoners join in the assertion, not
only that every observed change of motion has had a
cause, but that every change of motion must have a
cause. Here we have certain portions of substantial
and undoubted knowledge. Now the essential point in
the view which we must take of the idea of cause is
this, — that our view must be such as to form a solid
basis for our knowledge. We have, in the* Mechanical
Sciences, certain universal and necessary truths on the
subject of causes. Now any view which refers our be
lief in causation to mere experience or habit, cannot
explain the possibility of such necessary truths, since
experience and habit can never lead to a perception of
necessary connexion. But a view which teaches us to
acknowledge axioms concerning cause, as we acknow-
170 PHILOSOPHY OF THK MECHANICAL SCIENCES.
ledge axioms concerning space, will lead us to look upon
the science of mechanics as equally certain and univer
sal with the science of geometry ; and will thus mate
rially affect our judgment concerning the nature and
claims of our scientific knowledge.
Axioms concerning Cause, or concerning Force,
which as we shall see, is a modification of Cause, will
flow from an Idea of Cause, just as axioms concerning
space and number flow from the ideas of space and num
ber or time. And thus the propositions which con
stitute the science of Mechanics prove that we possess
an idea of cause, in the same sense in which the propo
sitions of geometry and arithmetic prove our possession
of the ideas of space and of time or number.
6. The idea of cause, like the ideas of space and
time, is a part of the active powers of the mind. The
relation of cause and effect is a relation or condition
under which events are apprehended, which relation is
not given by observation, but supplied by the mind itself.
According to the views which explain our apprehension
of cause by reference to habit, or to a supposed law of
our mental nature, causal connexion is a consequence of
agencies which the mind passively obeys ; but according
to the view to which we are led, this connexion is a
result of faculties which the mind actively exercises.
And thus the relation of cause and effect is a condition
of our apprehending successive events, a part of the
mind's constant and universal activity, a source of neces
sary truths ; or, to sum all this in one phrase, a Funda
mental Idea.
17
CHAPTER IV.
OF THE AXIOMS WHICH RELATE TO THE IDEA
OF CAUSE.
1 . Causes are abstract Conceptions. — WE have now
to express, as well as we can, the fundamental character
of that Idea of Cause, of which we have just proved the
existence. This may be done, at least for purposes of
reasoning, in this as in former instances, by means of
axioms. I shall state the principal axioms which belong
to this subject, referring the reader to his own thoughts
for the axiomatic evidence which belongs to them.
But I must first observe, that in order to express
general and abstract truths concerning cause and effect,
these terms, cause and effect, must be understood in a
general and abstract manner. When one event gives rise
to another, the first event is, in common language, often
called the cause, and the second the effect. Thus the
meeting; of two billiard balls may be said to be the
w
cause of one of them turning aside out of the path in
which it was moving. For our present purposes, how
ever, we must not apply the term cause to such occur
rences as this meeting and turning, but to a certain
conception, force, abstracted from all such special events,
and considered as a quality or property by which one
body affects the motion of the other. And in like man
ner in other cases, cause is_to be conceived as some
abstract quality, power, or efficacy, by which change is
produced ; a quality not identical \\itli the rvniK but
disclosed by means of them. Not only is this abstract
mode of conceiving force and cause useful in expressing
the fundamental principles of science ; but it supplies us
with the only mode by which such principles can be
• VOL. i. w. p. N
178 PHILOSOPHY OF THE MECHANICAL SCIENCES.
stated in a general manner, and made to lead to sub
stantial truth and real knowledge.
Understanding cause, therefore, in this sense, we
proceed to our Axioms.
2. First Axiom. Nothing can take place without a
Cause.
Every event, of whatever kind, must have a Cause in
the sense of the term which we have just indicated; and
that it must, is a universal and necessary proposition to
which we irresistibly assent as soon as it is understood.
We believe each appearance to come into existence, —
we conceive every change to take place, — not only with
something preceding it, but something by which it is made
to be what it is. An effect without a cause ; — an event
without a preceding condition involving the efficacy by
which the event is produced; — are suppositions which we
cannot for a moment admit. That the connexion of effect
with cause is universal and necessary, is a universal and
constant conviction of mankind. It persists in the minds
of all men, undisturbed by all the assaults of sophistry
and skepticism; and, as we have seen in the last chapter,
remains unshaken, even when its foundations seem to be
ruined. This axiom expresses, to a certain extent, our
Idea of Cause ; and when that idea is clearly appre
hended, the axiom requires no proof, and indeed admits
of none which makes it more evident. That notwith
standing its simplicity, it is of use in our speculations, we
shall hereafter see ; but in the first place, we must con
sider the other axioms belonging to this subject.
3. Second Axiom. Effects are proportional to their
Causes., and Causes are measured ~by their Effects.
We have already said that cause is that quality or
power, in the circumstances of each case, by which the
effect is produced ; and this power, an abstract property
of the condition of things to which it belongs, can in
AXIOMS wincii RELATE TO THE IDEA OF CAUSE. 1 79
no way fall directly under the cognizance of the senses.
Cause, of whatever kind, is not apprehended as including
objects and events which share its nature by being co-ex
tensive with certain portions of it, as space and time are.
It cannot therefore, like them, bo measured by repeti
tion of its own parts, as space is measured by repetition
of inches, and time by repetition of minutes. Causes may
be greater or less ; as, for instance, the force of a man is
greater than the force of a child. But how much is the
one greater than the other ? How are we to compare
the abstract conception, force, in such cases as these ?
To this, the obvious and only answer is, that we must
compare causes by means of their effects ; — that we must
compare force by something which force can do. The
child can lift one fagot; the man can lift ten such fagots:
we have here a means of comparison. And whether or
not the rule is to be applied in this manner, that is, by
the number of the things operated on, (a question which
we shall have to consider hereafter,) it is clear that this
form of rule, namely, a reference to some effect or other
as our measure, is the right, because the only possible
form. The cause determines the effect. The cause being
the same, the effect must be the same. The connexion
of the two is governed by a fixed and inviolable rule.
It admits of no ambiguity. Every degree of intensity
in the cause has some peculiar modification of the effect
corresponding to it. Hence the effect is an unfailing
index of the amount of the cause ; and if it be a mea
surable effect, gives a measure of the cause. We can
have no other measure ; but we need no other, for this
is exact, sufficient, and complete.
It may be said, that various effects are produced by
the same cause. The sun's heat melts wax and expands
quicksilver. The force of gravity causes bodies to move
downwards if they are free, and to press down upon their
N2
180 PHILOSOPHY OF Tilt: MECHANICAL SOIENCKS.
supports if they are supported. Which of the effects is to
be taken as the measure of heat, or of gravity, in these
cases ? To this we reply, that if we had merely different
states of the same cause to compare, any of the effects
might be taken. The sun's heat on different days might
be measured by the expansion of quicksilver, or by the
quantity of wax melted. The force of gravity, if it were
different at different places, might be measured by the
spaces through which a given weight would bend an
elastic support, or by the spaces through which a body
would fall in a given time. All these measures are con
sistent with the general character of our idea of cause.
4. Limitation of the Second Axiom. — But there may
be circumstances in the nature of the case which may
further determine the kind of effect which we must take
for the measure of the cause. For example, if causes
are conceived to be of such a nature as to be capable of
addition, the effects taken as their measure must conform
to this condition. This is the case with mechanical
causes. The weights of two bodies are the causes of the
pressure which they exert downwards ; and these weights
are capable of addition. The weight of the two is the
sum of the weight of each. We are therefore not at
liberty to say that weights shall be measured by the
spaces through which they bend a certain elastic support,
except we have first ascertained that the whole weight
bends it through a space equal to the sum of the inflec
tions produced by the separate weights. Without this
precaution, we might obtain inconsistent results. Two
weights, each of the magnitude 0 as measured by their
effects, might, if we took the inflections of a spring for
the effects, be together equal to 5 or to 7 by the same
kind of measurement. For the inflection produced by
two weights of 3 might, for aught we can see before
hand, be more or less than twice as great as the inflection
AXIOMS WHICH RELATE TO THE IDEA OK CAUSE. 181
produced by one weight of 3. That forces are capable of
addition, is a condition which limits, and, as we shall see,
in some cases rigorously fixes, the kind of effects which
are to be taken as their measures.
Causes which are thus capable of addition are to be
measured by the repeated addition of equal quantities.
Two such causes are equal to each other when they pro
duce exactly the same effect. So far our axiom is applied
directly. But these two causes can be added together ;
and being thus added, they are double of one of them ;
a*nd the cause composed by addition of three such, is
three times as great as the first ; and so on for any mea
sure whatever. By this means, and by this means only,
we have a complete and consistent measure of those
causes which are so conceived as to be subject to this
condition of being added and multiplied.
Causes are, in the present chapter, to be understood
in the widest sense of the term ; and the axiom now
under our consideration applies to them, whenever they
are of such a nature as to admit of any measure at all.
But the cases which we have more particularly in view
are mechanical causes, the causes of the motion and of
the equilibrium of bodies. In these cases, forces are con
ceived as capable of addition ; and what has been said of
the measure of causes in such cases, applies peculiarly to
mechanical forces. Two weights, placed together, may
be considered as a single weight, equal to the sum of the
two. Two pressures, pushing a body in the same direc
tion at the same point, are identical in all respects with
some single pressure, their SUM, pushing in like manner;
and this is true whether or not they put the body in
motion. In the cases of mechanical forces, therefore, we
take some certain effect, velocity generated or weight
supported, which may fix the unit offeree: and we then
measure all other forces by the successive repetition of
182 PHILOSOPHY OF THE MECHANICAL SCIENCES.
this unit, as we measure all spaces by the successive
repetition of our unit of lineal measure.
But these steps in the formation of the science of
Mechanics will be further explained, when we come to
follow our axioms concerning cause into their application
in that science. At present we have, perhaps, suffi
ciently explained the axiom that causes are measured
by their effects, and we now proceed to a third axiom,
also of great importance.
5. Third Axiom. Reaction is equal and opposite to
Action.
In the case of mechanical forces, the action of a
cause often takes place by an operation of one body
upon another ; and in this case, the action is always and
inevitably accompanied by an opposite action. If I press
a stone with my hand, the stone presses my hand in
return. If one ball strike another and put it in motion,
the second ball diminishes the motion of the first. In
these cases the operation is mutual ; the Action is ac
companied by a Reaction. And in all such cases the
Reaction is a force of exactly the same nature as the
Action, exerted in an opposite direction. A pressure
exerted upon a body at rest is resisted and balanced by
another pressure ; when the pressure of one body puts
another in motion, the body, though it yields to the force,
nevertheless exerts upon the pressing body a force like
that which it suffers.
Now the axiom asserts further, that this Reaction
is equal, as well as opposite, to the Action. For the
Reaction is an effect of the Action, and is determined by
it. And since the two, Action and Reaction, are forces
of the same nature, each may be considered as cause
and as effect ; and they must, therefore, determine each
other by a common rule. But this consideration leads
necessarily to their equality : for since the rule is mutual,
AXIOMS WHICH RELATE TO THE IDEA OF CAUSE.
if we could for an instant suppose the Reaction to be
less than the Action, we must, by the same rule, sup
pose the Action to be less than the Reaction. And thus
Action and Reaction, in every such case, are rigorously
equal to each other.
It is easily seen that this axiom is not a proposition
which is, or can be, proved by experience ; but that its
truth is anterior to special observation, and depends on
our conception of Action and Reaction. Like our other
axioms, this has its source in an Idea ; namely, the Idea
of Cause, under that particular condition in which cause
and effect are mutual. The necessary and universal
truth which we cannot help ascribing to the axiom, shows
that it is not derived from the stores of experience,
which can never contain truths of this character. Ac
cordingly, it was asserted with equal confidence and
generality by those who did not refer to experience for
their principles, and by those who did. Leonicus Tomreus,
a commentator of Aristotle, whose work was published
in 1552, and therefore at a period when no right opinions
concerning mechanical reaction were current, at least
in his school, says, in his remarks on the Author's Ques
tions concerning the communication of motion, that
"Reaction is equal and contrary to Action." The same
principle was taken for granted by all parties, in all the
controversies concerning the proper measure of force, of
which we shall have to speak : and would be rigorously
true, as a law of motion, whichever of the rival inter
pretations of the measure of the term ''Action" we were
to take.
6. Extent of the Third Axiom. — It may naturally be
asked whether this third Axiom respecting causation
extends to any other cases than those of mechanical
action, since the notion of Cause in general has certainly
a much wider extent. For instance, when a hot bodv
184 PHILOSOPHY OF THE MECHANICAL SCIENCES.
heats a cold one, is there necessarily an equal reaction
of the second body upon the first? Does the snowball
cool the boy's hand exactly as much as the hand heats
the snow ? To this we reply, that, in every case in which
one body acts upon another by its physical qualities, there
must be some reaction. No body can affect another
without being itself also affected. But in any physical
change the action exerted is an abstract term which may
be variously understood. The hot hand may melt a
cold body, or may warm it : which kind of effect is to
be taken as action? This remains to be determined by
other considerations.
In all cases of physical change produced by one body
in another, it is generally possible to assume such a
meaning of action, that the reaction shall be of the same
nature as the action ; and when this is done, the third
axiom of causation, that reaction is equal to action, is
universally true. Thus if a hot body heat a cold one,
the change may be conceived as the transfer of a certain
substance, heat or caloric, from the first body to the
second. On this supposition, the first body loses just as
much heat as the other gains ; action and reaction are
equal. But if the reaction be of a different kind to the
action we can no longer apply the axiom. If a hot body
melt a cold one, the latter cools the former : here, then, is
reaction ; but so long as the action and reaction are stated
in this form, we cannot assert any equality between them.
In treating of the secondary mechanical sciences, we
shall see further in what way we may conceive the
physical action of one body upon another, so that the
same axioms which are the basis of the science of
Mechanics shall apply to changes not at first sight mani
festly mechanical.
The three axioms of causation which we have now
stated are the fundamental maxims of all reasoning con-
AXIOMS WHICH RELATE TO THE IDEA OF CAUSE. 185
cerning causes as to their quantities; and it will be
shown in the sequel that these axioms form the basis of
the science of Mechanics, determining its form, extent,
and certainty. We must, however, in the first place,
consider how we acquire those conceptions upon which
the axioms now established are to be employed.
CHAPTER V.
OF THE ORIGIN OF OUR CONCEPTIONS OF
FORCE AND MATTER.
1. Force. — WHEN the faculties of observation and
thought are developed in man, the idea of causation is
applied to those changes which we see and feel in the
state of rest and motion of bodies around us. And
when our abstract conceptions are thus formed and
named, we adopt the term Force, and use it to
denote that property which is the cause of motion pro
duced, changed, or prevented. This conception is, it
would seem, mainly and primarily suggested by our
consciousness of the exertions by which we put bodies
in motion. The Latin and Greek words for Force, Vis,
FJV, were probably, like all abstract terms, derived at
first from some sensible object. The original meaning
of the Greek word was a muscle or tendon. Its first
application as an abstract term is accordingly to muscu
lar force.
aJr' Amc iro\v /ueioi/« Aaai/ «ei^«<;
rfK t*7riC(i/>;'(Tac, eTrepeifff (e FIN direXfflpov.
Then Ajax a far heavier stone upheaved,
He whirled it, and impressing Force intense
Upon the mass, dismist it.
The property by which bodies affect each other's
motions, was naturally likened to that energy which we
186 PHILOSOPHY OF THE MECHANICAL SCIENCES.
exert upon them with similar effect : and thus the labour
ing horse, the rushing torrent, the descending weight, the
elastic bow, were said to exert force. Homer* speaks
of the force of the river, R'? Trorano'io ; and Hesiodf of
the >force of the north wind, F<s dve^ov fiopeao.
Thus man's general notion of force was probably first
suggested by his muscular exertions, that is, by an act
depending upon that muscular sense, to which, as we
have already seen, the perception of space is mainly due.
And this being the case, it will be easily understood that
the Direction of the force thus exerted is perceived by
the muscular sense, at the same time that the force itself
is perceived ; and that the direction of any other force is
understood by comparison with force which man must
exert to produce the same effect, in the same manner as
force itself is so understood.
This abstract notion of Force long remained in a very
vague and obscure condition, as may be seen by referring
to the History for the failures of attempts at a science of
force and motion, made by the ancients and their com
mentators in the middle ages. By degrees, in modern
times, we see the scientific faculty revive. The concep
tion of Force becomes so far distinct and precise that it
can be reasoned upon in a consistent manner, with de
monstrated consequences ; and a genuine science of Me
chanics conies into existence. The foundations of this
science are to be found in the Axioms concerning causa
tion which we have already stated ; these axioms being
interpreted and fixed in their application by a constant
reference to observed facts, as we shall show. But we
must, in the first place, consider further those primary
processes of observation by which we acquire the first
materials of thought on such subjects.
2. Matter. — The conception of Force, as we have said,
* //. xxr. t Op. ct D.
ORIGIN OF CONCEPTIONS OF FORCE AND MATTER. 187
arises with our consciousness of our own muscular exer
tions. But we cannot imagine such exertions without
also imagining some bodily substance against which they
are exercised. If we press, we press something : if we
thrust or throw, there must be something to resist the
thrust or to receive the impulse. Without body, mus
cular force cannot be exerted and force in general is not
conceivable.
Thus Force cannot exist without Body on which it
acts. The two conceptions, Force and Matter, are co
existent and correlative. Force implies resistance ; and
the force is effective only wheiL the resistance Js called
into j)lay_. If we grasp a stone, we have no hold of it
till the closing of the hand is resisted by the solid tex
ture of the stone. If we push open a gate, we must
surmount the opposition which it exerts while turning
on its hinges. However slight the resistance be, there
must be some resistance, or there would be no force.
If we imagine a state of things in which objects do not
resist our touch, they must also cease to be influenced
by our strength. Such a state of things we sometimes
imagine in our dreams; and such are the poetical pic
tures of the regions inhabited by disembodied spirits. In
these, the figures which appear are conspicuous to the
eye, but impalpable like shadow or smoke ; and as they
do not resist the corporeal impressions, so neither do
they obey them. The spectator tries in vain to strike
or to grasp them.
Et ni cana vates tcnues sine corpore vitas
A dm ono.it volitare c.iva sub imagine forma1,
Irruat ac frustra ferro diverbcret umbras.
The Sibyl warns him that there round him fly
Bodiless things, but substance to the eye ;
Else had he pierced those shapes with life-like face,
And smitten, fierce, the unresisting space.
188 PHILOSOPHY OF THE MECHANICAL SCIENCES.
Neque ilium
Prensantem nequicquam umbras et multa volentem
Diccrc, preterca vidit.
He grasps her form, and clutches but the shade.
Such may be the circumstances of the unreal world of
dreams, or of poetical fancies approaching to dreams :
for in these worlds our imaginary perceptions are bound
by no rigid conditions of force and reaction. In such
cases, the mind casts off the empire of the idea of cause,
as it casts off even the still more familiar sway of the
ideas of space and time. But the character of the
material world in which we live when awake is, that we
have at every instant and at every place, force operating
on matter and matter resisting force.
3. Solidity. — From our consciousness of muscular
exertion, we derive, as we have seen, the conception of
force, and with that also the conception of matter. We
have already shown, in a former chapter, that the same
part of our frame, the muscular system, is the organ by
which we perceive extension and the relations of space.
Thus the same organ gives us the perception of body as
resisting force, and as occupying space ; and by combin
ing these conditions we have the conception of solid
extended bodies. In reality, this resistance is inevitably
presented to our notice in the very facts from which we
collect the notion of extension. For the action of the
hand and arm by which we follow the forms of objects,
implies that we apply our fingers to their surface ; and
we are stopped there by the resistance which the body
offers. This resistance is precisely that which is requisite
in order to make us conscious of our muscular effort*.
Neither touch, nor any other mere passive sensation,
could produce the perception of extent, as we have
already urged : nor could the muscular sense lead to such
* Brown's Lectures, i. 46fi.
ORIGIN OF CONCEPTIONS OF FORCE AND MATTER. 189
a perception, except the extension of the muscles were
felt to be resisted. And thus the perception of resistance
enters the mind along with the perception of extended
bodies. All the objects with which we have to do are
not only extended but solid.
This sense of the term solidity, (the general property
of all matter,) is different to that in which we oppose
stolidity to Jiuidity. We may avoid ambiguity by op
posing rigid to fluid bodies. By solid bodies, as we now
speak of them, we mean only such as resist the pressure
which we exert, so long as their parts continue in their
places. By fluid bodies, we mean those whose parts are,
by a slight pressure, removed out of their places. A drop
of water ceases to prevent the contact of our two hands,
not by ceasing to have solidity in this sense, but by being
thrust out of the way. If it could remain in its place,
it could not cease to exercise its resistance to our pres
sure, except by ceasing to be matter altogether.
The perception of solidity, like the perception of
extension, implies an act of the mind, as well as an
impression of the senses : as the perception of extension
implies the idea of space, so the perception of solidity
implies the idea of action and reaction. That an Idea
is involved in our knowledge on this subject appears, as
in other instances, from this consideration, that the con
victions of persons, even of those who allow of no ground
of knowledge but experience, do in fact go far beyond the
possible limits of experience. Thus Locke says*, that
" the bodies which we daily handle hinder by an insur
mountable force the approach of the parts of our hands
that press them." Now it is manifest that our observa
tion can never go to this length. By our senses we can
only perceive that bodies resist the greatest actual forces
that we exert upon them. But our conception of force
* E&say, B. ii. c. 4.
190 PHILOSOPHY OF THE MECHANICAL SCIENCES.
carries us further : and since, so long as the body is
there to receive the action of the force, it must suffer
the whole of that action, and must react as much as
it suffers : it is therefore true, that so long as the body
remains there, the force which is exerted upon it can
never surmount the resistance which the body exercises.
And thus this doctrine, that bodies resist the intrusion
of other bodies by an insurmountable force, is, in fact,
a consequence of the axiom that the reaction is always
equal to the action.
4. Inertia. — But this principle of the equality of
action and reaction appears also in another way. Not
only when we exert force upon bodies at rest, but when,
by our exertions, we put them in motion, they react. If
we set a large stone in motion, the stone resists ; for the
operation requires an effort. By increasing the effort, we
can increase the effect, that is, the motion produced ; but
the resistance still remains. And the greater the stone
moved, the greater is the effort requisite to move it.
There is, in every case, a resistance to motion, which shows
itself, not in preventing the motion, but in a reciprocal
force, exerted backwards upon the agent by which the
motion is produced. And this resistance resides in
each portion of matter, for it is increased as we add
one portion of matter to another. We can push a light
boat rapidly through the water ; but we may go on
increasing its freight, till we are barely able to stir it.
This property of matter, then, by which it resists the
reception of motion, or rather by which it reacts and
requires an adequate force in order that any motion may
result, is called its inertness, or inertia. That matter has
such a property, is a conviction flowing from that idea of
a reaction equal and opposite to the action, which the
conception of all force involves. By what laws this
inertia depends on the magnitude, form, and material of
ORIGIN OF CONCEPTIONS OF FORCE AND MATTER. 191
the body, must be the subject of our consideration here
after. But that matter has this inertia, in virtue of
which, as the matter is greater, the velocity which the
same effort can communicate to it is less, is a principle
inseparable from the notion of matter itself.
Hermann says that Kepler first introduced this " most
significant word" inertia. Whether it is to be found in
earlier writers I know not ; Kepler certainly does use it
familiarly in those attempts to assign physical reasons
for the motions of the planets which were among the
main occasions of the discovery of the true laws of me
chanics. He assumes the slowness of the motions of the
planets to increase, (other causes remaining the same,)
as the inertia increases ; and though, even in this as
sumption, there is an errour involved, (if we adopt that
interpretation of the term inertia to which subsequent
researches led,) the introduction of such a word was one
step in determining and expressing those laws of motion
which depend on the fundamental principle of the equality
of action and reaction.
5. We have thus seen, I trust in a satisfactory
manner, the origin of our conceptions of Force, Matter,
Solidity, and Inertness. It has appeared that the organ
by which we obtain such conceptions is that very mus
cular frame, which is the main instrument of our percep
tions of space ; but that, besides bodily sensations, these
ideal conceptions, like all the others which we have
hitherto considered, involve also an habitual activity of
the mind, giving to our sensations a meaning which they
could not otherwise possess. And among the ideas thus
brought into play, is an idea of action with an equal and
opposite reaction, which forms a foundation for univer
sal truths to be hereafter established respecting the
conceptions thus obtained.
We must now endeavour to trace in what manner
192 PHILOSOPHY OF THE MECHANICAL SCIENCES.
these fundamental principles and conceptions are un
folded by means of observation and reasoning, till they
become an extensive yet indisputable science.
CHAPTER VI.
OF THE ESTABLISHMENT OF THE PRINCIPLES
OF STATICS.
1. Object of the Chapter. — IN the present and the
succeeding chapters we have to show how the general
axioms of Causation enable us to construct the science
of Mechanics. We have to consider these axioms as
moulding themselves, in the first place, into certain fun
damental mechanical principles, which are of evident
and necessary truth in virtue of their dependence upon
the general axioms of Causation ; and thus as forming a
foundation for the whole structure of the science ; — a
system of truths no less necessary than the fundamen
tal principles, because derived from these by rigorous
demonstration.
This account of the construction of the science of
Mechanics, however generally treated, cannot be other
wise than technical in its details, and will probably be
imperfectly understood by any one not acquainted with
Mechanics as a mathematical science.
I cannot omit this portion of my survey without
rendering my work incomplete ; but I may remark that
the main purpose of it is to prove, in a more particular
manner, what I have already declared in general, that
there are, in Mechanics no less than in Geometry, funda
mental principles of axiomatic evidence and necessity;
— that these principles derive their axiomatic character
from the Idea which they involve, namely the Idea of
ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 193
Cause ; — and that through the combination of principles
of this kind, the whole science of Mechanics, including
its most complex and remote results, exists as a body of
solid and universal truths.
2. Statics and Dynamics. — We must first turn our
attention to a technical distinction of Mechanics into
two portions, according as the forces about which we
reason produce rest, or motion ; the former portion is
termed Statics, the latter Dynamics. If a stone fall,
or a weight put a machine in motion, the problem
belongs to Dynamics; but if the stone rest upon the
ground, or a weight be merely supported by a machine,
without being raised higher, the question is one of
Statics.
3. Equilibrium. — In Statics, forces balance each
other, or keep each other in equilibrium. And forces
which directly balance each other, or keep each other in
equilibrium, are necessarily and manifestly equal. If
we see two boys pull at two ends of a rope so that
neither of them in the smallest degree prevails over the
other, we have a case in which two forces are in equili
brium. The two forces are evidently equal, and are a
statical exemplification of action and reaction, such as are
spoken of in the third axiom concerning causes. Now
the same exemplification occurs in every case of equili
brium. No point or body can be kept at rest except in
virtue of opposing forces acting upon it ; and these forces
must always be equal in their opposite effect. When a
stone lies on the floor, the weight of the stone down
wards is opposed and balanced by an equal pressure of
the floor upwards. If the stone rests on a slope, its
tendency to slide is counteracted by some equal and
opposite force, arising, it may be, from the resistance
which the sloping ground opposes to any motion along
its surface. Every case of rest is a case of equilibrium :
VOL. i. \v. P. 0
194 PHILOSOPHY OF THE MECHANICAL SCIENCES.
every case of equilibrium is a case of equal and opposite
forces.
The most complex frame-work on which weights are
supported, as the roof of a building, or the cordage of a
machine, are still examples of equilibrium. In such
cases we may have many forces all combining to balance
each other ; and the equilibrium will depend on various
conditions of direction and magnitude among the forces.
And in order to understand what are these conditions,
we must ask, in the first place, what we understand by
the magnitude of such forces ; — what is the measure of
statical forces.
4. Measure of Statical Forces. — At first we might
expect, perhaps, that since statical forces come under the
general notion of Cause, the mode of measuring them
would be derived from the second axiom of Causation,
that causes are measured by their effects. But we find
that the application of this axiom is controlled by the
limitation which we noticed, after stating that axiom ;
namely, the condition that the causes shall be capable of
addition. Further, as we have seen, a statical force pro
duces no other effect than this, that it balances some
other statical force ; and hence the measure of statical
forces is necessarily dependent upon their balancing,
that is, upon the equality of action and reaction.
That statical forces are capable of addition is involved
in our conception of such forces. When two men pull
at a rope in the same direction, the forces which they
exert are added together. When two heavy bodies are
put into a basket suspended by a string, their weights
are added, and the sum is supported by the string.
Combining these considerations, it will appear that
the measure of statical forces is necessarily given at once
by the fundamental principle of the equality of action
and reaction. Since two opposite forces which balance
ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 195
oach other are equal, each force is measured by that
which it balances ; and since forces are capable of addi
tion, a force of any magnitude is measured by adding to
gether a proper number of such equal forces. Thus a
heavy body which, appended to some certain elastic
branch of a tree, would bend it down through one inch,
•may be taken as a unit of weight. Then if we remove
this first body, and find a second heavy body which will
also bend the branch through the same space, this is also
a unit of weight ; and in like manner we might go on to
a third and a fourth equal body ; and adding together
the two, or the three, or the four heavy bodies, we have
a force twice, or three times, or four times the unit of
•weight. And with such a collection of heavy bodies, or
weights, we can readily measure all other forces ; for the
same principle of the equality of action and reaction
leads at once to this maxim, that any statical force is
measured by the weight which it would support.
As has been said, it might at first have been sup
posed that we should have to apply, in this case, the
axiom that causes are measured by their effects in an
other manner; that thus, if that body were a unit of
weight which bent the bough of a tree through one inch,
that body would be two units which bent it through two
inches, and so on. But, as we have already stated, the
measures of weight must be subject to this condition,
that they are susceptible of being added : and therefore
we cannot take the deflexion of the bough for our mea
sure, till we have ascertained, that which experience
alone can teach us, that under the burden of two equal
weights, the deflexion will be twice as great as it is with
one weight, which is not true, or at least is neither ob
viously nor necessarily true. In this, as in all other cases,
although causes must be measured by their effects, we
learn from experience only how the effects are to be
196 PHILOSOPHY OF THE MECHANICAL SCIENCES.
interpreted, so as to give a true and consistent mea
sure.
With regard, however, to the measure of statical
force, and of weight, no difficulty really occurred to phi
losophers from the time when they first began to specu
late on such subjects ; for it was easily seen that if we
take any uniform material, as wood, or stone, or iron,
portions of this which are geometrically equal, must also
be equal in statical effect ; since this was implied in the
very hypothesis of a uniform material. And a body ten
times as large as another of the same substance, will be
of ten times the weight. But before men could esta
blish by reasoning the conditions under which weights
would be in equilibrium, some other principles were
needed in addition to the mere measure of forces. The
principles introduced for this purpose still resulted from
the conception of equal action and reaction ; but it re
quired no small clearness of thought to select them
rightly, and to employ them successfully. This, however,
was done, to a certain extent, by the Greeks ; and the
treatise of Archimedes On the Center of Gravity, is
founded on principles which may still be considered as
the genuine basis of statical reasoning. I shall make a
few remarks on the most important principle among
those which Archimedes thus employs.
5. The Center of Gravity. — The most important of
the principles which enter into the demonstration of
Archimedes is this : that " Every body has a center of
gravity ;" meaning by the center of gravity, a point at
which the whole matter of the body may be supposed to
be collected, to all intents and purposes of statical
reasoning. This principle has been put in various forms
by succeeding writers : for instance, it has been thought
sufficient to assume a case much simpler than the general
one; and to assert that two equal bodies have their
ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 197
center of gravity in the point midway between them. It
is to be observed, that this assertion not only implies
that the two bodies will balance upon a support placed
at that midway point, but also, that they will exercise,
upon such a support, a pressure equal to their sum ;
for this point being the center of gravity, the whole
matter of the two bodies may be conceived to be col
lected there, and therefore the whole weight will press
there. And thus the principle in question amounts to
this, that when two equal heavy bodies are supported on
the middle point between them, the pressure upon the
support is equal to the sum of the weights of the bodies.
A clear understanding of the nature and grounds of
this principle is of great consequence : for in it we have
the foundation of a large portion of the science of
Mechanics. And if this principle can be shown to be
necessarily true, in virtue of our Fundamental Ideas, we
can hardly doubt that there exist many other truths of
the same kind, and that no sound view of the evidence
and extent of human knowledge can be obtained, so long
as we mistake the nature of these, its first principles.
The above principle, that the pressure on the support
is equal to the sum of the bodies supported, is often
stated as an axiom in the outset of books on Mechanics.
And this appears to be the true place and character of
this principle, in accordance with the reasonings which
we have already urged. The axiom depends upon our
conception of action and reaction. That the two weights
are supported, implies that the supporting force must be
equal to the force or weight supported.
In order further to show the foundation of this
principle, we may ask the question : — If it be not an
axiom, deriving its truth from the fundamental concep
tion of equal action and reaction, which equilibrium
always implies, what is the origin of its certainty? The
198 PHILOSOPHY OF THE MECHANICAL SCIENCES.
principle is never for an instant denied or questioned: it is
taken for granted, even before it is stated. No one will
doubt that it is not only true, but true with the same
rigour and universality as the axioms of Geometry. Will
it be said, that it is borrowed from experience ? Expe
rience could never prove a principle to be universally
and rigorously true. Moreover, when from experience
we prove a proposition to possess great exactness and
generality, we approach by degrees to this proof: the
conviction becomes stronger, the truth more secure, as
we accumulate trials. But nothing of this kind is the
case in the instance before us. There is no gradation
from less to greater certainty ; — no hesitation which
precedes confidence. From the first, we know that the
axiom is exactly and certainly true. In order to be
convinced of it, we do not require many trials, but
merely a clear understanding of the assertion itself.
But in fact, not only are trials not necessary to the
proof, but they do not strengthen it. Probably no
one ever made a trial for the purpose of showing that
the pressure upon the support is equal to the sum of the
two weights. Certainly no person with clear mechanical
conceptions ever wanted such a trial to convince him of
the truth ; or thought the truth clearer after the trial
had been made. If to such a person, an experiment
were shown which seemed to contradict the principle, his
conclusion would be, not that the principle was doubtful,
but that the apparatus was out of order. Nothing can
be less like collecting truth from experience than this.
We maintain, then, that this equality of mechanical
action and reaction, is one of the principles which do
not flow from, but regulate our experience. To this
principle, the facts which we observe must conform ;
and we cannot help interpreting them in such a manner
that they shall be exemplifications of the principle. A
ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 199
mechanical pressure not accompanied by an equal and
opposite pressure, can no more be given by experience,
than two unequal right angles. With the supposition of
such inequalities, space ceases to be space, force ceases to
be force, matter ceases to be matter. And this equality
of action and reaction, considered in the case in which
two bodies are connected so as to act on a single support,
leads to the axiom which we have stated above, and
which is one of the main foundations of the science of
Mechanics.
6. Oblique Forces. — By the aid of this axiom and
a few others, the Greeks made some progress in the
science of Statics. But after a short advance, they
arrived at another difficulty, that of Oblique Forces,
which they never overcame; and which no mathematician
mastered till modern times. The unpublished manuscripts
of Leonardo da Vinci, written in the fifteenth century,
and the works of Stevinus and Galileo, in the sixteenth,
are the places in which we find the first solid grounds of
reasoning on the subject of forces acting obliquely to
each other. And mathematicians, having thus become
possessed of all the mechanical principles which are
requisite in problems respecting equilibrium, soon framed
a complete science of Statics. Succeeding writers pre
sented this science in forms variously modified ; for it
was found, in Mechanics as in Geometry, that various
propositions might be taken as the starting points ; and
that the collection of truths which it was the mecha
nician's business to include in his course, might thus be
traversed by various routes, each path offering a series
of satisfactory demonstrations. The fundamental con
ceptions of force and resistance, like those of space and
number, could be contemplated under different aspects,
each of which might be made the basis of axioms,
or of principles employed as axioms. Hence the
200 PHILOSOPHY OF THE MECHANICAL SCIENCES.
grounds of the truth of Statics may be stated in various
ways ; and it would be a task of some length to examine
all these completely, and to trace them to their Funda
mental Ideas. This I shall not undertake here to do ;
but the philosophical importance of the subject makes
it proper to offer a few remarks on some of the main
principles involved in the different modes of presenting
Statics as a rigorously demonstrated science.
7. A Force may be supposed to act at any Point of its
Direction. — It has been stated in the history of Mecha
nics*, that Leonardo da Vinci and Galileo obtained the
true measure of the effect of oblique forces, by reason
ings which were, in substance, the same. The principle
of these reasonings is that expressed at the head of this
paragraph ; and when we have a little accustomed our
selves to contemplate our conceptions of force, and its
action on matter, in an abstract manner, we shall have
no difficulty in assenting to the principle in this general
form. But it may, perhaps, be more obvious at first in
a special case.
If we suppose a wheel, moveable about its axis, and
carrying with it in its motion a weight, (as, for example,
one of the wheels by means of which the large bells of a
church are rung,) this weight may be supported by means
of a rope (not passing along the circumference of the
wheel, as is usual in the case of bells,) but fastened to
one of the spokes of the wheel. Now the principle which
is enunciated above asserts, that if the rope pass in a
straight line across several of the spokes of the wheel, it
makes no difference in the mechanical effect of the force
applied, for the purpose of putting the bell in motion, to
which of these spokes the rope is fastened. In each case,
the fastening of the rope to the wheel merely serves to
enable the force to produce motion about the centre ;
* Hist. Ind. ScL, B. vi. c. i. sect. 2. and Note (A).
ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 2()1
and so long as the force acts in the same line, the effect
is the same, at whatever point of the rope the line of
action finishes.
This axiom very readily aids us in estimating the
effect of oblique forces. For when a force acts on one of
the arms of a lever at any oblique angle, we suppose
another arm projecting from the centre of motion, like
another spoke of the same wheel, so situated that it is
perpendicular to the force. This arm we may, with
Leonardo, call the virtual lexer ; for, by the axiom, we
may suppose the force to act where the line of its direc
tion meets this arm ; and thus we reduce the case to
that in which the force acts perpendicularly on the arm.
The ground of this axiom is, that matter, in Statics,
is necessarily conceived as transmitting force. That force
can be transmitted from one place to another, by means
of matter ; — that we can push with a rod, pull with a
rope, — are suppositions implied in our conceptions of
force and matter. Matter is, as we have said, that which
receives the impression of force, and the modes just
mentioned, are the simplest ways in which that impres
sion operates. And since, in any of these cases, the force
might be resisted by a reaction equal to the force itself,
the reaction in each case would be equal, and, therefore,
the action in each case is necessarily equal ; and thus the
forces must be transmitted, from one point to another,
without increase or diminution.
This property of matter, of transmitting the action of
force, is of various kinds. We have the coherence of a
rope which enables us to pull, and the rigidity of a staftj
which enables us to push with it in the direction of its
length ; and again, the same staff has a rigidity of another
kind, in virtue of which we can use it as a lever ; that is, a
rigidity to resist flexure, and to transmit the force which
turns a body round a fulcrum. There is, further, the
202 PHILOSOPHY OF THE MECHANICAL SCIENCES.
rigidity by which a solid body resists twisting. Of these
kinds of rigidity, the first is that to which our axiom
refers ; but in order to complete the list of the ele
mentary principles of Statics, we ought also to lay down
axioms respecting the other kinds of rigidity*. These,
however, I shall not here state, as they do not involve
any new principle. Like the one just considered, they
form part of our fundamental conception of matter ; they
are not the results of any experience, but are the hypo
theses to which we are irresistibly led, when we would
liberate our reasonings concerning force and matter from
a dependence on the special results of experience. We
cannot even conceive (that is, if we have any clear
mechanical conceptions at all) the force exerted by the
point of a staff and resisting the force which we steadily
impress on the head of it, to be different from the
impressed force.
8. Forces may have equivalent Forces substituted for
them. The Parallelogram of Forces. — It has already been
observed, that in order to prove the doctrines of Statics,
we may take various principles as our starting points,
and may still find a course of demonstration by which
the leading propositions belonging to the subject may
be established. Thus, instead of beginning our reason
ings, as in the last section we supposed them to
commence, with the case in which forces act upon
different points of the same body in the same line of
force, and counteract each other in virtue of the inter
vening matter by which the effect of force is transferred
from one point to another, we may suppose different
forces to act at the same point, and may thus commence
our reasonings with a case in which we have to con
template force, without having to take into our account
* Such axioms are given in a little work (The Mechanical Euclid}
which I published on the Elements of Mechanics.
ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 203
the resistance or rigidity of matter. Two statical forces,
thus acting at a mathematical point, are equivalent, in
al 1 respects, to some single force acting at tjie same point ;
and would be kept in equilibrium by a force equal and
opposite to that single force. And the rule by which
the single force is derived from the two, is commonly
termed the parallelogram offerees; the proposition being
this, — That if the two forces be represented in magnitude
and direction by the two sides of a parallelogram, the
resulting force will be represented in the same manner
by the diagonal of the parallelogram. This proposition
has very frequently been made, by modern writers, the
commencement of the science of Mechanics : a position
for which, by its simplicity, it is well suited ; although,
in order to deduce from it the other elementary proposi
tions of the science, as, for instance, those respecting the
lever, we require the axiom stated in the last section.
9. The Parallelogram of Forces is a necessary Truth.
— In the series of discussions in which we are here
engaged, our main business is to ascertain the nature and
grounds of the certainty of scientific truths. We have,
therefore, to ask whether this proposition, the parallelo
gram of forces, be a necessary truth ; and if so, on what
grounds its necessity ultimately rests. We shall find
that this, like the other fundamental doctrines of Statics,
justly claims a demonstrative certainty. Daniel Ber
noulli, in 1726, gave the first proof of this important
proposition on pure statical principles ; and thus, as he
says*, "proved that statical theorems are not less
necessarily true than geometrical are." If we examine
this proof of Bernoulli, in order to discover what are
the principles on which it rests, we shall find that the
reasoning employs in its progress such axioms as this ; —
That if from forces which arc in equilibrium at a point
* Comm. Pctrop. Vol. i.
204 PHILOSOPHY OF THE MECHANICAL SCIENCES.
be taken away other forces which are in equilibrium at
the same point, the remainder will be in equilibrium ;
and generally ; — That if forces can be resolved into other
equivalent forces, these may be separated, grouped, and
recombined, in any new manner, and the result will still
be identical with what it was at first. Thus in Ber
noulli's proof, the two forces to be compounded are repre
sented by P and Q ; p is resolved into two other forces, x
and u ; and Q, into two others, Y and v, under certain
conditions. It is then assumed that these forces may be
grouped into the pairs x, Y, and u, v : and when it has
been shown that x and Y are in equilibrium, they may, by
what has been said, be removed, and the forces, p, Q, are
equivalent to u, v ; which, being in the same direction
by the course of the construction, have a result equal to
their sum.
It is clear that the principles here assumed are
genuine axioms, depending upon our conception of the
nature of equivalence of forces, and upon their being
capable of addition and composition. If the forces P, Q,
be equivalent to forces x, u, Y, v, they are equivalent to
these forces added and compounded in any order ; just
as a geometrical figure is, by our conception of space,
equivalent to its parts added together in any order. The
apprehension of forces as having magnitude, as made
up of parts, as capable of composition, leads to such
axioms in Statics, in the same manner as the like
apprehension of space leads to the axioms of Geometry.
And thus the truths of Statics, resting upon such founda
tions, are independent of experience in the same manner
in which geometrical truths are so.
The proof of the parallelogram of forces thus given
by Daniel Bernoulli, as it was the first, is also one of
the most simple proofs of that proposition which have
been devised up to the present day. Many other demon-
ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 205
strations, however, have been given of the same proposi
tion. Jacobi, a German mathematician, has collected
and examined eighteen of these"'. They all depend
either upon such principles as have just been stated ;
That forces may in every way be replaced by those which
are equivalent to them ; — or else upon those previously
stated, the doctrine of the lever, and the transfer of a
force from one point to another of its direction. In
either case, they are necessary results of our statical con
ceptions, independent of any observed laws of motion,
and indeed, of the conception of actual motion altogether.
There is another class of alleged proofs of the paral
lelogram of forces, which involve the consideration of
the motion produced by the forces. But such reasonings
are, in fact, altogether irrelevant to the subject of Statics.
In that science, forces are not measured by the motion
which they produce, but by the forces which they will
balance, as we have already seen. The combination of
two forces employed in producing motion in the same
body, either simultaneously or successively, belongs to
that part of Mechanics which has motion for its subject,
and is to be considered in treating of the laws of motion.
The composition of motion, (as when a man moves in a
ship while the ship moves through the water,) has con
stantly been confounded with the composition of force.
But though it has been done by very eminent mathe
maticians, it is quite necessary for us to keep the two
subjects distinct, in order to see the real nature of the
evidence of truth in either case. The conditions of equi
librium of two forces on a lever, or of three forces at
* These arc by the following mathematicians; D. Bernoulli
(1726); Lambert (1771); Scarella (175fi); Venini (1 704); Araldi
(180(>); Wachter (IHlii); Kajstner ; Marini ; Eytelwein ; Salimbeni ;
Duchayla; two different proofs by Foncenex (17GO) ; three by
D'Alembert; and those of Laplace and M. Poisson.
206 PHILOSOPHY OF THE MECHANICAL SCIENCES.
a point, can be established without any reference what
ever to any motions which the forces might, under other
circumstances, produce. And because this can be done,
to do so is the only scientific procedure. To prove such
propositions by any other course, would be to support
truth by extraneous and inconclusive reasons; which
would be foreign to our purpose, since we seek not only
knowledge, but the grounds of our knowledge.
10. The Center of gravity seeks the lowest place. —
The principles which we have already mentioned afford
a sufficient basis for the science of Statics in its most
extensive and varied applications ; and the conditions of
equilibrium of the most complex combinations of ma
chinery may be deduced from these principles with a
rigour not inferior to that of geometry. But in some of
the more complex cases, the results of long trains of
reasoning may be foreseen, in virtue of certain maxims
which appear to us self-evident, although it may not be
easy to trace the exact dependence of these maxims upon
our fundamental conceptions of force and matter. Of
this nature is the maxim now stated ; — That in any com
bination of matter any how supported, the Center of
Gravity will descend into the lowest position which the
connexion of the parts allows it to assume by descend
ing. It is easily seem that this maxim carries to a much
greater extent the principle which the Greek mathe
maticians assumed, that every body has a Center of
Gravity, that is, a point in which, if the whole matter of
the body be collected, the effect will remain unchanged.
For the Greeks asserted this of a single rigid mass only ;
whereas, in the maxim now under our notice, it is asserted
of any masses, connected by strings, rods, joints, or in
any manner. We have already seen that more modern
writers on mechanics, desirous of assuming as funda
mental no wider principles than are absolutely necessary,
ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 207
have not adopted the Greek axiom in all its generality,
but have only asserted that two equal weights have a
center of gravity midway between them. Yet the prin
ciple that every body, however irregular, has a center of
gravity, and will be supported if that center is supported,
and not otherwise, is so far evident, that it might be
employed as a fundamental truth, if we could not resolve
it into any simpler truths : and, historically speaking, it
was assumed as evident by the Greeks. In like manner
the still wider principle, that a collection of bodies, as,
for instance, a flexible chain hanging upon one or more
supports, has a center of gravity ; and that this point
will descend to the lowest possible situation, as a single
body would do, has been adopted at various periods in
the history of mechanics ; and especially at conjunctures
when mathematical philosophers have had new and dif
ficult problems to contend with. For in almost every
instance it has only been by repeated struggles that phi
losophers have reduced the solution of such problems to
a clear dependence upon the most simple axioms.
11. Stevinuss Proof for Oblique Forces. — We have
an example of this mode of dealing with problems, in
Stevinus's mode of reasoning concerning the Inclined
Plane ; which, as we have stated in the History of Me
chanics, was the first correct published solution of that
problem. Stevinus supposes a loop of chain, or a loop
of string loaded with a series of equal balls at equal dis
tances, to hang over the Inclined Plane ; and his reason
ing proceeds upon this assumption, — That such a loop
so hanging will find a certain position in which it will
rest : for otherwise, says he*, its motion must go on for
ever, which is absurd. It may be asked how this absurd
ity of a perpetual motion appears ; and it will perhaps
be added, that although the impossibility of a machine
* Stcvin. Statiqnc, Livrc I., prop. 19.
208 PHILOSOPHY OF THE MECHANICAL SCIENCES.
with such a condition may be proved as a remote result
of mechanical principles, this impossibility can hardly
be itself recognized as a self-evident truth. But to this
we may reply, that the impossibility is really evident in
the case contemplated by Stevinus ; for we cannot con
ceive a loop of chain to go on through all eternity, slid
ing round and round upon its support, by the effect of
its own weight. And the ground of our conviction that
this cannot be, seems to be this consideration; that when
the chain moves by the effect of its weight, we consider
its motion as the result of an effort to reach some certain
position, in which it can rest ; just as a single ball in
a bowl moves till it comes to rest at the lowest point
of the bowl. Such an effect of weight in the chain, we
may represent to ourselves by conceiving all the matter
of the chain to be collected in one single point, and this
single heavy point to hang from the support in some way
or other, so as fitly to represent the mode of support of
the chain. In whatever manner this heavy point (the
center of gravity of the chain) be supported and con
trolled in its movements, there will still be some position
of rest which it will seek and find. And thus there will
be some corresponding position of rest for the chain ; and
the interminable shifting from one position to another,
with no disposition to rest in any position, cannot exist.
Thus the demonstration of the property of the
Inclined Plane by Stevinus, depends upon a principle
which, though far from being the simplest of those to
which the case can be reduced, is still both true and
evident : and the evidence of this principle, depending
upon the assumption of a center of gravity, is of the
same nature as the evidence of the Greek statical demon
strations, the earliest real advances in the science.
12. Principle of Virtual Velocities. — We have
referred above to an assertion often made, that we
ESTABLISHMENT OF THE PRINCIPLES OF STATICS 209
may, from the simple principles of Mechanics, demon
strate the impossibility of a perpetual motion. In reality,
however, the simplest proof of that impossibility, in
a machine acted upon by weight only, arises from the
very maxim above stated, that the center of gravity seeks
and finds the lowest place ; or from some similar propo
sition. For if, as is done by many writers, we profess
to prove the impossibility of a perpetual motion by means
of that proposition which includes the conditions of equi
librium, and is called the Principle of Virtual Velocities*,
we are under the necessity of first proving in a general
manner that principle. And if this be done by a mere
enumeration of cases, (as by taking those five cases which
are called the Mechanical Powers,} there may remain
some doubts whether the enumeration of possible mecha
nical combinations be complete. Accordingly, some writers
have attempted independent and general proofs of the
Principle of Virtual Velocities; and these proofs rest
upon assumptions of the same nature as that now under
notice. This is, for example, the case with Lagrange's
proof, which depends upon what he calls the Principle
of Pulleys. For this principle is, — That a weight any
how supported, as by a string passing round any number
of pulleys any how placed, will be at rest then only,
when it cannot get lower by any small motion of the
pulleys. And thus the maxim that a weight will descend
if it can, is assumed as the basis of this proof.
There is, as we have said, no need to assume such
principles as these for the foundation of our mechanical
science. But it is, on various accounts, useful to direct
our attention to those cases in which truths, apprehended
at first in a complex and derivative form, have after
wards been reduced to their simpler elements ; — in which,
also, sagacious and inventive men have fixed upon those
* See Hist. I»d. Sci., B. vi. c. ii. sect. 4.
VOL. I. \V. P. I'
210 PHILOSOPHY OF THE MECHANICAL SCIENCES.
truths as self-evident, which now appear to us only cer
tain in virtue of demonstration. In these cases we can
hardly doubt that such men were led to assert the
doctrines which they discovered, not by any capricious
conjecture or arbitrary selection, but by having a keener
and deeper insight than other persons into the relations
which were the object of their contemplation ; and in the
science now spoken of, they were led to their assump
tions by possessing clearly and distinctly the conceptions
of mechanical cause and effect, — action and reaction, —
force, and the nature of its operation.
13. Fluids 2^'ess Equally in all Directions. — The
doctrines which concern the equilibrium of fluids depend
on principles no less certain and simple than those which
refer to the equilibrium of solid bodies ; and the Greeks,
who, as we have seen, obtained a clear view of some of
the principles of Statics, also made a beginning in the
kindred subject of Hydrostatics. We still possess a trea
tise of Archimedes On Floating Bodies, which contains
correct solutions of several problems belonging to this
subject, and of some which are by no means easy. In
this treatise, the fundamental assumption is of this kind :
"Let it be assumed that the nature of a fluid is such,
that the parts which are less pressed yield to those which
are more pressed." In this assumption or axiom it is
implied that a pressure exerted upon a fluid in one direc
tion produces a pressure in another direction ; thus, the
weight of the fluid which arises from a downward force
produces a lateral pressure against the sides of the con
taining vessel. Not only does the pressure thus diverge
from its original direction into all other directions, but the
pressure, is in all directions exactly equal, an equal extent
of the fluid being taken. This principle, which was in
volved in the reasoning of Archimedes, is still to the
present day the basis of all hydrostatical treatises, and is
ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 211
expressed, as above, by saying that fluids press equally
in all directions.
Concerning this, as concerning previously-noticed
principles, we have to ask whether it can rightly be said
to be derived from experience. And to this the answer
must still be, as in the former cases, that the proposition
is not one borrowed from experience in any usual or
exact sense of the phrase. I will endeavour to illustrate
this. There are many elementary propositions in phy
sics, our knowledge of which indisputably depends upon
experience ; and in these cases there is no difficulty in
seeing the evidence of this dependence. In such cases,
the experiments which prove the law are prominently
stated in treatises upon the subject : they are given with
exact measures, and with an account of the means by
which errors were avoided : the experiments of more
recent times have either rendered more certain the law
originally asserted, or have pointed out some correction
of it as requisite : and the names, both of the discoverers
of the law and of its subsequent reformers, are well
known. For instance, the proposition that " The elastic
force of air varies as the density," was first proved by
Boyle, by means of operations of which the detail is given
in his Defence of his Pneumatical Experiments* ; and
by Marriotte in his Traite de VEquilibre des Liquides,
from whom it has generally been termed Marriotte's law.
After being confirmed by many other experimenters,
this law was suspected to be slightly inaccurate, and a
commission of the French Academy of Sciences was
;i|>j)ointed, consisting of several distinguished philoso-
phersf, to ascertain the truth or falsehood of this suspicion.
* Shaw's Boyle, Vol. ii. p. 671-
t The members were Prony, Arago, Ampere, Girard, and Dulong.
The experiments were extended to a pressure of twenty-seven atmo-
•pherea ; and in no instance did the difference between the observed
P 2
212 PHILOSOPHY OF THE MECHANICAL SCIENCES.
The result of their investigations appeared to be, that
the law is exact, as nearly as the inevitable inaccuracies
of machinery and measures will allow us to judge. Here
we have an example of a law which is of the simplest
kind and form ; and which yet is not allowed to rest
upon its simplicity or apparent probability, but is rigor
ously tested by experience. In this case, the assertion,
that the law depends upon experience, contains a refer
ence to plain and notorious passages in the history of
science.
Now with regard to the principle that fluids press
equally in all directions, the case is altogether different.
It is, indeed, often asserted in works on hydrostatics,
that the principle is collected from experience, and some
times a few experiments are described as exhibiting its
effect ; but these are such as to illustrate and explain,
rather than to prove, the truth of the principle : they
are never related to have been made with that exact
ness of precaution and measurement, or that frequency
of repetition, which are necessary to establish a purely
experimental truth. Nor did such experiments occur as
important steps in the history of science. It does not
appear that Archimedes thought experiment necessary
to confirm the truth of the law as he employed it : on
the contrary, he states it in exactly the same shape as
the axioms which he employs in statics, and even in geo
metry ; namely, as an assumption. Nor does any intel
ligent student of the subject find any difficulty in assent
ing to this fundamental principle of hydrostatics as soon
as it is propounded to him. Experiment was not requi
site for its discovery; experiment is not necessary for
its proof at present ; and we may add, that experiment,
and calculated elasticity amount to one-hundredth of the whole ; nor
did the difference appear to increase with the increase of pressure. —
Fechner, Repertorium, i. 110.
ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 213
though it may make the proposition more readily intelli
gible, can add nothing to our conviction of its truth
when it is once understood.
14. Foundation of the abore Axiom. — But it will
naturally be asked, What then is the ground of our
conviction of this doctrine of the equal pressure of a
fluid in all directions'? And to this I reply, that the
reasons of this conviction are involved in our idea of a
fluid, which is considered as matter, and therefore as
capable of receiving, resisting, and transmitting force
according to the general conception of matter; and which
is also considered as matter which has its parts perfectly
movcable among one another. For it follows from
these suppositions, that if the fluid be confined, a pres
sure which thrusts in one side of the containing vessel,
may cause any other side to bulge outwards, if there be
a part of the surface which has not strength to resist
this pressure from within. And that this pressure, when
thus transferred into a direction different from the ori
ginal one, is not altered in intensity, depends upon this
consideration ; that any difference in the two pressures
would be considered as a defect of jwrfect fluidity, since
the fluidity would be still more complete, if this entire
and undiminished transmission of pressure in all direc
tions were supposed. If, for instance, the lateral pres
sure were less than the vertical, this could be conceived
no other way than as indicating some rigidity or adhesion
of the parts of the fluid. When the fluidity is perfect,
the two pressures which act in the two different parts of
the fluid exactly balance each other : they are the action
and the reaction; and must hence be equal by the same
iH-cessity as two directly opposite forces in statics.
But it may be urged, that even if we grant that this
<•< inception of a perfect fluid, as a body which has its
parts perfectly moveable among each other, leads us
214 PHILOSOPHY OF THE MECHANICAL SCIENCES.
necessarily to the principle of the equality of hydrostatic
pressure in all directions, still this conception itself is
obtained from experience, or suggested by observation.
And to this we may reply, that the conception of a fluid,
as contemplated in mechanical theory, cannot be said to
be derived from experience, except in the same manner
as the conception of a solid and rigid body may be said
to be acquired by experience. For if we imagine a
vessel full of small, smooth spherical balls, such a collec
tion of balls would approach to the nature of a fluid, in
having its parts moveable among each other ; and would
approach to perfect fluidity, as the balls became
smoother and smaller. And such a collection of balls
would also possess the statical properties of a fluid ; for
it would transmit pressure out of a vertical into a lateral
(or any other) direction, in the same manner as a fluid
would do. And thus a collection of solid bodies has
the same property which a fluid has ; and the science
of Hydrostatics borrows from experience no principles
beyond those which are involved in the science of
Statics respecting solids. And since in this latter por
tion of science, as we have already seen, none of the
principles depend for their evidence upon any special
experience, the doctrines of Hydrostatics also are not
proved by experience, but have a necessary truth bor
rowed from the relations of our ideas.
It is hardly to be expected that the above reasoning
will, at first sight, produce conviction in the mind of the
reader, except he have, to a certain extent, acquainted
himself with the elementary doctrines of the science of
Hydrostatics as usually delivered ; and have followed,
with clear and steady apprehension, some of the trains
of reasoning by which the pressures of fluids are deter
mined; as, for instance, the explanation of what is called
the Hydrostatic Parndo.r, The necessity of such a dis-
ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 215
cipline in order that the reader may enter fully into this
part of our speculations, naturally renders them less
popular ; but this disadvantage is inevitable in our plan.
We cannot expect to throw light upon philosophy by
means of the advances which have been made in the
mathematical and physical sciences, except we really
understand the doctrines which have been firmly esta
blished in those sciences. This preparation for philoso
phizing may be somewhat laborious ; but such labour is
necessary if we would pursue speculative truth with all
the advantages which the present condition of human
knowledge places within our reach.
We may add, that the consequences to which we arc
directed by the preceding opinions, are of very great im
portance in their bearing upon our general views respect
ing human knowledge. I trust to be able to show, that
some important distinctions arc illustrated, some per
plexing paradoxes solved, and some large anticipations
of the future extension of our knowledge suggested, by
means of the conclusions to which the preceding discus
sions have conducted us. But before I proceed to these
general topics, I must consider the foundations of some
of the remaining portions of Mechanics.
CHAPTER VII.
OF THE ESTABLISHMENT OF THE PRINCIPLES
OF DYNAMICS.
1. IN the History of Mechanics, I have traced the
steps by which the three Laws of Motion and the other
principles of mechanics were discovered, established, and
extended to the widest generality of form and applica
tion. We have, in these laws, examples of principles
which were, historically speaking, obtained by reference
216 PHILOSOPHY OF THE MECHANICAL SCIENCES.
to experience. Bearing in mind the object and the re
sult of the preceding discussions, we cannot but turn
with much interest to examine these portions of science ;
to inquire whether there be any real difference in the
grounds and nature between the knowledge thus ob
tained, and those truths which we have already contem
plated ; and which, as we have seen, contain their own
evidence, and do not require proof from experiment.
2. The First Lam of Motion. — The first law of mo
tion is, that When a body moves not acted upon by any
force, it mill go on perpetually in a straight line., and
with a uniform Telocity. Now what is the real ground
of our assent to this proposition ? That it is not at first
sight a self-evident truth, appears to be clear ; since from
the time of Aristotle to that of Galileo the opposite
assertion was held to be true ; and it was believed that
all bodies in motion had, by their own nature, a constant
tendency to move more and more slowly, so as to stop at
last. This belief, indeed, is probably even now enter
tained by most persons, till their attention is fixed upon
the arguments by which the first law of motion is esta
blished. It is, however, not difficult to lead any person
of a speculative habit of thought to see that the retard
ation which constantly takes place in the motion of all
bodies when left to themselves, is, in reality, the effect
of extraneous forces which destroy the velocity. A top
ceases to spin because the friction against the ground
and the resistance of the air gradually diminish its mo
tion, and not because its motion has any internal prin
ciple of decay or fatigue. This may be shown, and was,
in fact, shown by Hooke before the Royal Society, at the
time when the laws of motion were still under discus
sion, by means of experiments in which the weight of
the top is increased, and the resistance to motion offered
by its support, is diminished ; for by such contrivances
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 217
> .•
its motion is made to continue much longer than it
would otherwise do. And by experiments of this nature,
although we can never remove the whole of the external
impediments to continued motion, and although, conse
quently, there will always be some retardation ; and an
end of the motion of a body left to itself, however long
it may be delayed, must at last come ; yet we can esta
blish a conviction that if all resistance could de removed,
there would be no diminution of velocity, and thus the
motion would go on for ever.
If we call to mind the axioms which we formerly
stated, as containing the most important conditions
involved in the idea of Cause, it will be seen that our
conviction in this case depends upon the first axiom of
Causation, that nothing can happen without a cause.
Every change in the velocity of the moving body must
have a cause ; and if the change can, in any manner, be
referred to the presence of other bodies, these are said
to exert force upon the moving body: and the conception
of force is thus evolved from the general idea of cause.
Force is any cause which has motion, or change of
motion, for its effect ; and thus, all the change of velocity
of a body which can be referred to extraneous bodies, — as
the air which surrounds it, or the support on which it
rests, — is considered as the effect of forces; and this
consideration is looked upon as explaining the difference
between the motion which really takes places in the expe
riment, and that motion which, as the law asserts, would
take place if the body were not acted on by any forces.
Thus the truth of the first law of motion depends
upon the axiom that no change can take place without a
cause; and follows from the definition of force, if we sup
pose that there can be none but an external cause of
change. But in order to establish the law, it was neces
sary further to be assured that tlr re is no internal cause
218 PHILOSOPHY OF THE MECHANICAL SCIENCES.
of change of velocity belonging to all matter whatever,
and operating in such a manner that the mere progress
of time is sufficient to produce a diminution of velocity
in all moving bodies. It appears from the history of
mechanical science, that this latter step required a refer
ence to observation and experiment ; and that the first
law of motion is so far, historically at least, dependent
upon our experience.
But notwithstanding this historical evidence of the
need which we have of a reference to observed facts, in
order to place this first law of motion out of doubt, it has
been maintained by very eminent mathematicians and
philosophers, that the law is, in truth, evident of itself,
and does not really rest upon experimental proof. Such,
for example, is the opinion of D'Alembert*, who offers
what is called an d priori proof of this law ; that is, a
demonstration derived from our ideas alone. When a
body is put in motion, either, he says, the cause which
puts it in motion at first, suffices to make it move one
foot, or the continued action of the cause during this foot
is requisite for the motion. In the first case, the same
reason which made the body proceed to the end of the
first foot will hold for its going on through a second,
a third, a fourth foot, and so on for any number. In
the second case, the same reason which made the force
continue to act during the first foot, will hold for its
acting, and therefore for the body moving during each
succeeding foot. And thus the body, once beginning to
move, must go on moving for ever.
It is obvious that we might reply to this argument,
that the reasons for the body proceeding during each
succeeding foot may not necessarily be all the same ; for
among these reasons may be the time which has elapsed;
and thus the velocity may undergo a change as the time
ESTABLISHMENT OF TUE PRINCIPLES OF DYNAMICS. 211)
proceeds : and we require observation to inform us that
it does not do so.
Professor Playfair has presented nearly the same
argument, although in a different and more mathematical
form*. If the velocity change, says he, it must change
according to some expression of calculation depending
upon the time, or, in mathematical language, must be a
function of the time. If the velocity diminish as the
time increases, this may be expressed by stating the velo
city in each case as a certain number, from which another
quantity, or term, increasing as the time increases, is
subtracted. But, Playfair adds, there is no condition
involved in the nature of the case, by which the coeffi
cients, or numbers which are to be employed, along with
the number representing the time, in calculating this
second term, can be determined to be of one magnitude
rather than of any other. Therefore he infers there can
be no such coefficients, and that the velocity is in each
case equal to some constant number, independent of the
time ; and is therefore the same for all times.
In reply to this we may observe, that the circum
stance of our not seeing in the nature of the case any
thing which determines for us the coefficients above-
spoken off, cannot prove that they have not some certain
value in nature. We do not see in the nature of the
case anything which should determine a body to fall six
teen feet in a second of time, rather than one foot or one
hundred feet : yet in fact the space thus run through by
falling bodies is determined to a certain magnitude. It
would be easy to assign a mathematical expression for
the velocity of a body, implying that one-hundredth of the
velocity, or any other fraction, is lost in each second f:
* Outlines, &c., j>. 2(5.
+ This would IK; the case, if, / being the number of seconds elapsed,
220 PHILOSOPHY OF THE MECHANICAL SCIENCES.
and where is the absurdity of supposing such an expres
sion really to represent the velocity ?
Most modern writers on mechanics have embraced
the opposite opinion, and have ascribed our knowledge
of this first law of motion to experience. Thus M.
Poisson, one of the most eminent of the mathematicians
who have written on this subject, says*, " We cannot
affirm a priori that the velocity communicated to a body
will not become slower and slower of itself, and end by
being entirely extinguished. It is only by experience
and induction that this question can be decided."
Yet it cannot be denied that there is much force in
those arguments by which it is attempted to shew that
the First Law of Motion, such as we find it, is more
consonant to our conceptions than any other would be.
The Law, as it exists, is the most simple that we can
conceive. Instead of having to determine by experi
ments what is the law of the natural change of velocity,
we find the Law to be that it does not change at all. To a
certain extent, the Law depends upon the evident axiom,
that no change can take place without a cause. But
the question further occurs, whether the mere lapse of
time may not be a cause of change of velocity. In order
to ensure this, we have recourse to experiment ; and the
result is that time alone does not produce any such
change. In addition to the conditions of change which
we collect from our own Ideas, we ask of Experience what
other conditions and circumstances she has to offer ; and
the answer is, that she can point out none. When we
have removed the alterations which external causes, in
and C some constant quantity, the velocity were expressed by this
mathematical formula,
Poisson, Dynamique, Ed. 2, Art. 113.
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. '2'2l
our very conception of them, occasion, there are no
longer any alterations. Instead of having to guide our
selves by experience, we learn that on this subject she
has nothing to tell us. Instead of having to take into
account a number of circumstances, we find that we have
only to reject all circumstances. The velocity of a body
remains unaltered by time alone, of whatever kind the
body itself be.
But the doctrine that time alone is not a cause of
change of velocity in any body is further recommended
to us by this consideration ; — that time is conceived by
us not as a cause, but only as a condition of other causes
producing their effects. Causes operate in time ; but it
is only when the cause exists, that the lapse of time can
give rise to alterations. When therefore all external
causes of change of velocity are supposed to be removed,
the velocity must continue identical with itself, whatever
the time which elapses. An eternity of negation can
produce no positive result.
Thus, though the discovery of the First Law of
Motion was made, historically speaking, by means of
experiment, we have now attained a point of view in
which we see that it might have been certainly known
to be true independently of experience. This law in its
ultimate form, when completely simplified and steadily
contemplated, assumes the character of a self-evident
truth. We shall find the same process to take place in
other instances. And this feature in the progress of
science will hereafter be found to suggest very important
views with regard both to the nature and prospects of
our knowledge.
3. Gravity is a Uniform Force. — We shall find
observations of the same kind offering themselves in a
manner more or less obvious, with regard to the other
principles of Dynamics. The determination of the laws
'2'2'2 PHILOSOPHY OF THE MECHANICAL SCIENCES.
according to which bodies fall downwards by the com
mon action of gravity, has already been noticed in the
History of Mechanics*, as one of the earliest positive
advances in the doctrine of motion. These laws were
first rightly stated by Galileo, and established by rea
soning and by experiment, not without dissent and con
troversy. The amount of these doctrines is this : That
gravity is a uniform accelerating force ; such a uniform
force having this for its character, that it makes the
velocity increase in exact proportion to the time of
motion. The relation which the spaces described by the
body bear to the times in which they are described, is
obtained by mathematical deduction from this definition
of the force.
The clear Definition of a uniform accelerating force,
and the Proposition that gravity is such a force, were
co-ordinate and contemporary steps in this discovery.
In defining accelerating force, reference, tacit or ex
press, was necessarily made to the second of the general
axioms respecting causation, — That causes are measured
by their effects. Force, in the cases now under our
notice, is conceived to be, as we have already stated,
(p. 217,) any cause which, acting from without, changes
the motion of a body. It must, therefore, in this accep
tation, be measured by the magnitude of the changes
which are produced. But in what manner the changes
of motion are to be employed as the measures of force, is
learnt from observation of the facts which we see taking
place in the world. Experience interprets the axiom of
causation, from which otherwise we could not deduce
any real knowledge. We may assume, in virtue of our
general conceptions of force, that under the same cir
cumstances, a greater change of motion implies a greater
force producing it ; but what are we to expect when the
* Hist. hid. Sci., B vi. c. ii. sect. 2.
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 2'23
circumstances change ? The weight of a body makes it
fall from rest at first, and causes it to move more quickly
as it descends lower. We may express this by saying,
that gravity, the universal force which makes all terres
trial bodies fall when not supported, by its continuous
action first gives velocity to the body when it has none,
and afterwards adds velocity to that which the body
already has. But how is the velocity added propor
tioned to the velocity which already exists? Force
acting on a body at rest, and on a body in motion,
appears under very different conditions ; — how are the
effects related ? Let the force be conceived to be in both
cases the same, since force is conceived to depend upon
the extraneous bodies, and not upon the condition of the
moving mass itself. But the force being the same, the
effects may still be different. It is at first sight con
ceivable that the body, acted upon by the same gravity,
may receive a less addition of velocity when it is already
moving in the direction in which this gravity impels it ;
for if we ourselves push a body forwards, we can produce
little additional effect upon it when it is already moving
rapidly away from us. May it not be true, in like man
ner, that although gravity be always the same force, its
effect depends upon the velocity which the body under
its influence already possesses ?
Observation and reasoning combined, as we have
said, enabled Galileo to answer these questions. He as
serted and proved that we may consistently and properly
measure a force by the velocity which is by it generated
in a body, in some certain time, as one second ; and
further, that if we adopt this measure, gravity will be a
force of the same value under all circumstances of the
body which it affects; since it appeared that, in fact, a
falling body does receive equal increments of velocity
in equal times from first to last.
'2'24 PHILOSOPHY OF THE MECHANICAL SCIENCES.
If it be asked whether we could have known, anterior
to, or independent of, experiment, that gravity is a uni
form force in the sense thus imposed upon the term ;
it appears clear that we must reply, that we could not
have attained to such knowledge, since other laws of the
motion of bodies downwards are easily conceivable, and
nothing but observation could inform us that one of
these laws does not prevail in fact. Indeed, we may add,
that the assertion that the force of gravity is uniform, is
so far from being self-evident, that it is not even true ;
for gravity varies according to the distance from the
center of the earth ; and although this variation is so
small as to be, in the case of falling bodies, imperceptible,
it negatives the rigorous uniformity of the force as com
pletely, though not to the same extent, as if the weight
of a body diminished in a marked degree, when it was
carried from the lower to the upper room of a house. It
cannot, then, be a truth independent of experience, that
gravity is uniform.
Yet, in fact, the assertion that gravity is uniform was
assented to, not only before it was proved, but even
before it was clearly understood. It was readily granted
by all, that bodies which fall freely are uniformly accele
rated ; but while some held the opinion just stated, that
uniformly accelerated motion is that in which the velocity
increases in proportion to the time, others maintained,
that that is uniformly accelerated motion, in which the
velocity increases in proportion to the space ; so that, for
example, a body in falling vertically through twenty feet
should acquire twice as great a velocity as one which
falls through ten feet.
These two opinions are both put forward by the
interlocutors of Galileo's Dialogue on this subject*. And
the latter supposition is rejected, the author showing,
* DialogOj IIT. p. 95.
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 225
not that it is inconsistent with experience, but that it is
impossible in itself: inasmuch as it would inevitably lead
to the conclusion, that the fall through a large and a
small vertical space would occupy exactly the same time.
Indeed, Galileo assumes his definition of uniformly
accelerated motion as one which is sufficiently recom
mended by its own simplicity. " If we attend carefully,"
he says, "we shall find that no mode of increase of velocity
is more simple than that which adds equal increments in
equal times. Which we may easily understand if we
consider the close affinity of time and motion : for as the
uniformity of motion is defined by the equality of spaces
described in equal times, so we may conceive the uni
formity of acceleration to exist when equal velocities are
added in equal times."
Galileo's mode of supporting his opinion, that bodies
falling by the action of gravity are thus uniformly acce
lerated, consists, in the first place, in adducing the
maxim that nature always employs the most simple
means*. But he is far from considering this a decisive
argument. " I," says one of his speakers, " as it would
be very unreasonable in me to gainsay this or any other
definition which any author may please to make, since
they are all arbitrary, may still, without offence, doubt
whether such a definition, conceived and admitted in the
abstract, fits, agrees, and is verified in that kind of
accelerated motion which bodies have when they descend
naturally."
The experimental proof that bodies, when they fall
downwards, are uniformly accelerated, is (by Galileo)
derived from the inclined plane ; and therefore assumes
the proposition, that if such uniform acceleration prevail
in vertical motion, it will also hold when a body is com
pelled to describe an oblique rectilinear path. This pro-
* Dialogo, m. j>. 91.
VOL. I. W. T. Q
226 PHILOSOPHY OF THE MECHANICAL SCIENCES.
position may be shown to be true, if (assuming by anti
cipation the Third Law of Motion, of which we shall
shortly have to speak,) we introduce the conception of
a uniform statical force as the cause of uniform acce
leration. For the force on the inclined plane bears
a constant proportion to the vertical force, and this
proportion is known from statical considerations. But
in the work of which we are speaking, Galileo does
not introduce this abstract conception of force as the
foundation of his doctrines. Instead of this, he pro
poses, as a postulate sufficiently evident to be made
the basis of his reasonings, That bodies which descend
down inclined planes of different inclinations, but of
the same vertical height, all acquire the same velocity"".
But when this postulate has been propounded by one
of the persons of the dialogue, another interlocutor says,
" You discourse very probably ; but besides this like
lihood, I wish to augment the probability so far, that
it shall be almost as complete as a necessary demon
stration." He then proceeds to describe a very inge
nious and simple experiment, which shows that when a
body is made to swing upwards at the end of a string,
it attains to the same height, whatever is the path it
follows, so long as it starts from the lowest point with
the same velocity. And thus Galileo's postulate is ex
perimentally confirmed, so far as the force of gravity can
be taken as an example of the forces which the postulate
contemplates : and conversely, gravity is proved to be a
uniform force, so far as it can be considered clear that
the postulate is true of uniform forces.
When we have introduced the conception and defi
nition of accelerating force, Galileo's postulate, that
bodies descending down inclined planes of the same
vertical height, acquire the same velocity, may, by a
* Dialogo, in. p. 36.
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 227
few steps of reasoning, be demonstrated to be true of
uniform forces : and thus the proof that gravity, either in
vertical or oblique motion, is a uniform force, is con
firmed by the experiment above mentioned ; as it also is,
on like grounds, by many other experiments, made upon
inclined planes and pendulums.
Thus the propriety of Galileo's conception of a uni
form force, and the doctrine that gravity is a uniform
force, were confirmed by the same reasonings and experi
ments. We may make here two remarks ; First, that the
conception, when established and rightly stated, appears
so simple as hardly to require experimental proof; a
remark which we have already made with regard to the
First Law of Motion : and Second, that the discovery of
the real law of nature was made by assuming proposi
tions which, without further proof, we should consider as
very precarious, and as far less obvious, as well as less
evident, than the law of nature in its simple form.
4. The Second Law of Motion. — When a body, instead
of falling downwards from rest, is thrown in any direc
tion, it describes a curve line, till its motion is stopped.
In this, and in all other cases in which a body describes
a curved path in free space, its motion is determined by
the Second Law of Motion. The law, in its general
form, is as follows : — When a body is thus cast forth
and acted upon by a force in a direction transverse to its
motion, the result is, That there is combined with the
motion with which the body is thrown, another motion,
exactly the same as that which the same force would have
<-(>iinnunicated to a body at rest.
It will readily be understood that the basis of this
law is the axiom already stated, that effects are measured
by their causes. In virtue of this axiom, the effect of
gravity acting upon a body in a direction transverse to its
motion, must measure the accelerative or deflective force
2*28 PHILOSOPHY OF THK MECHANICAL SCIENCES.
of gravity under those circumstances. If this effect vary
with the varying velocity and direction of the body thus
acted upon, the deflective force of gravity also will vary
with those circumstances. The more simple supposition
is, that the deflective force of gravity is the same, whatever
be the velocity and direction of the body which is sub
jected to its influence : and this is the supposition which
we find to be verified by facts. For example, a ball let
fall from the top of a ship's upright mast, when she is
sailing steadily forward, will fall at the foot of the mast,
just as if it were let fall while the ship were at rest ; thus
showing that the motion which gravity gives to the ball
is compounded with the horizontal motion which the ball
shares with the ship from the first. This general and
simple conception of motions as compounded with one
another, represents, it is proved, the manner in which
the motion produced by gravity modifies any other mo
tion which the body may previously have had.
The discussions which terminated in the general re
ception of this Second Law of Motion among mechanical
writers, were much mixed up with the arguments for and
against the Copernican system, which system represented
the earth as revolving upon its axis. For the obvious
argument against this system was, that if each point of the
earth's surface were thus in motion from west to east, a
stone dropt from the top of a tower would be left behind,
the tower moving away from it : and the answer was, that
by this law of motion, the stone would have the earth's
motion impressed upon it, as well as that motion which
would arise from its gravity to the earth ; and that the
motion of the stone relative to the tower would thus be
the same as if both earth and tower were at rest. Gali
leo further urged, as a presumption in favour of the opi
nion that the two motions, — the circular motion arising
from the rotation of the earth, and the downward motion
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 220
arising from the gravity of the stone, would be com
pounded in the way we have described, (neither of them
disturbing or diminishing the other,) — that the first
motion was in its own nature not liable to any change or
diminution*, as we learn from the First Law of Motion.
Nor was the subject lightly dismissed. The experiment
of the stone let fall from the top of the mast was made
in various forms by^'Gassendi ; and in his Epistle, De
Motu impresso a Motore translate, the rule now in ques
tion is supported by reference to these experiments. In
this manner, the general truth, the Second Law of
Motion, was established completely and beyond dispute.
But when this law had been proved to be true in a
general sense, with such accuracy as rude experiments,
like those of Galileo and Gassendi, would admit, it still
remained to be ascertained (supposing our knowledge of
the law to be the result of experience alone,) whether it
were true with that precise and rigorous exactness which
more refined modes of experimenting could test. We
so willingly believe in the simplicity of laws of nature,
that the rigorous accuracy of such a law, known to be at
least approximately true, was taken for granted, till some
ground for suspecting the contrary should appear. Yet
calculations have not been wanting which might confirm
the law as true to the last degree of accuracy. Laplace
relates (Syst. du Monde, livre iv., chap. 1C,) that at one
time he had conceived it possible that the effect of
gravity upon the moon might be slightly modified by the
moon's direction and velocity ; and that in this way an
explanation might be found for the moon's deceleration
(a deviation of her observed from her calculated place,
which long perplexed mathematicians). But it was after
some time discovered that this feature in the moon's
motion arose from another cause; and the second law of
* I)id/o<fu, n. ji. 11-1.
230 PHILOSOPHY OF THE MECHANICAL SCIENCES.
motion was confirmed as true in the most rigorous
sense.
Thus we see that although there were arguments
which might be urged in favour of this law, founded
upon the necessary relations of ideas, men became con
vinced of its truth only when it was verified and con
firmed by actual experiment. But yet in this case
again, as in the former ones, when the law had been
established beyond doubt or question, men were very
ready to believe that it was not a mere result of observa
tion, — that the truth which it contained was not derived
from experience, — that it might have been assumed as
true in virtue of reasonings anterior to experience, — and
that experiments served only to make the law more plain
and intelligible, as visible diagrams in geometry serve to
illustrate geometrical truths ; our knowledge not being
(they deemed) in mechanics, any more than in geometry,
borrowed from the senses. It was thought by many to
be self-evident, that the effect of a force in any direction
cannot be increased or diminished by any motion trans
verse to the direction of the force which the body may
have at the same time : or, to express it otherwise, that
if the motion of the body be compounded of a horizontal
and vertical motion, the vertical motion alone will be
affected by the vertical force. This principle, indeed,
not only has appeared evident to many persons, but even
at the present day is assumed as an axiom by many of
the most eminent mathematicians. It is, for example,
so employed in the Mccanique Celeste of Laplace, which
may be looked upon as the standard of mathematical
mechanics in our time; and in the Mecanique Analy-
tique of Lagrange, the most consummate example which
has appeared of subtilty of thought on such subjects, as
well as of power of mathematical generalization*. And
* I may observe that the rule that we may compound motions, as
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 231
thus we have here another example of that circumstance
which we have already noticed in speaking of the First
Law of Motion, (Art. 2 of this Chapter,) and of the Law
that Gravity is a uniform Force, (Art. 3) ; namely, that
the law, though historically established by experiments,
appears, when once discovered and reduced to its most
simple and general form, to be self-evident. I am the
more desirous of drawing attention to this feature in
various portions of the history of science, inasmuch as it
will be found to lead to some very extensive and impor
tant views, hereafter to be considered.
5. The Third Law of Motion. — We have, in the
definition of Accelerating Force, a measure of Forces, so
far as they are concerned in producing motion. We had
before, in speaking of the principles of statics, defined
the measure of Forces or Pressures, so far as they are
employed in producing equilibrium. But these two
aspects of Force are closely connected; and we require a
law which shall lay down the rule of their connexion.
By the same kind of muscular exertion by which we
the Law supposes, is involved in the step of resolving them ; which is
done in the passage to which I refer (Mcc. Analyt. Ptie. I., sect. i. art. 3,
p. 225). " Si on concoit que la mouvement d'un corps et les forces
qui le sollicitent soient decomposes suivant trois lignes droites perpen-
diculaires entre elles, on pourra considerer separement les mouvemens
et les forces relatives a rhacun a de ces trois directions. Car a cause de
la pi-rpendicularite dcs directions il est visible que chacun de ces mouve-
meiis partiels peut etre regarde comme independant des deux autres,
et qu'il ne peut recevoir d'alteration que de la part de la force qui agit
dans la direction de ce mouvement ; Ton pent conclure que ces trois
inouvements doivent suivre, chacun en particulier, les lois des mouve-
tiiriis rectilignes acceleres ou retardes par les forces donnees." Laplace
makes the same assumption in effect, (Mcc Cel. P. I., liv. i., art. 7»)
hy resolving the forces which act upon a point in three rectangular
din-rtimis, and reasoning separately concerning each direction. But iu
lii> mode of treating the subject is involved a principle which belongs
to the Third Law of Motion, namely, the doctrine that the velocity is
;i- the force, of which we shall have to speak elsewhere.
232 PHILOSOPHY OF THE MECHANICAL SCIENCES.
can support a heavy stone, we can also put it in motion.
The question then occurs, how is the rate and manner
of its motion determined ? The answer to this question
is contained in the Third Law of Motion, and it is to
this effect : that the Momentum which any pressure pro
duces in the mass in a given time is proportional to the
pressure. By Momentum is meant the product of the
numbers which express the velocity and the mass of the
body : and hence, if the mass of the body be the same
in the instances which we compare, the rule is, — That
the Telocity is as the force which produces it; and this is
one of the simplest ways of expressing the Third Law
of Motion.
In agreement with our general plan, we have to ask.
What is the ground of this rule ? What is the simplest
and most satisfactory form to which we can reduce the
proof of it ? Or, to take an instance ; if a double pres
sure be exerted against a given mass, so disposed as to
be capable of motion, why must it produce twice the
velocity in the same time ?
To answer this question, suppose the double pressure
to be resolved into two single pressures : one of these
will produce a certain velocity; and the question is, why
an equal pressure, acting upon the same mass, will pro
duce an equal velocity in addition to the former? Or,
stating the matter otherwise, the question is, why each
of the two forces will produce its separate effect, unal
tered by the simultaneous action of the other force ?
This statement of the case makes it seem to approach
very near to such cases as are included in the Second
Law of Motion, and therefore it might appear that this
Third Law has no grounds distinct from the Second.
But it must be recollected that the \\ordforce has a dif
ferent meaning in this case and in that ; in this place it
signifies pressure; in the statement of the Second Law
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 233
its import was accelerative or deflective force, measured
by the velocity or deflexion generated. And thus the
Third Law of Motion, so far as our reasonings yet go,
appears to rest on a foundation different from the Second.
Accordingly, that part of tho Third Law of Motion
which we are now considering, that the velocity gene
rated is as the force, was obtained, in fact, by a separate
train of research. The first exemplification of this law
which was studied by mathematicians, was the motion
of bodies upon inclined planes : for the force which urges
a body down an inclined plane is known by statics, and
hence the velocity of its descent was to be determined.
Galileo originally* in his attempts to solve this problem
of the descent of a body down an inclined plane, did not
proceed from the principle which we have stated, (the
determination of the force which acts down the inclined
plane from statical considerations,) obvious as it may
seem ; but assumed, as we have already seen, a propo
sition apparently far more precarious ; — namely, that
a body sliding down a smooth inclined plane acquires
always the same velocity, so long as the vertical height
fallen through is the same. And this conjecture, (for
at first it was nothing more than a conjecture,) he
confirmed by an ingenious experiment ; in which bodies
acquired or lost the same velocity by descending or
ascending through the same height, although their paths
were different in other respects.
This was the form in which the doctrine of the mo
tion of bodies down inclined planes was at first presented
in Galileo's Dialogues on the Science of Motion. But
his disciple Viviani was dissatisfied with the assumption
thus introduced ; and in succeeding editions of the Dia
logues, the apparent chasm in the reasoning was much
narrowed, by making the proof depend upon a principle
* /)/'«/. delta Sc. NHOI<. in., p. '.)(). Sec Hist. Ind. Sci. R vi. c. ii. (sect. 5.
234 PHILOSOPHY OF THE MECHANICAL SCIENCES.
nearly identical with the third law of motion as we have
just stated it. In the proof thus added, " We are agreed,"
says the interlocutor"", "that in a moving body the
impetus, energy, momentum, or propension to motion, is
as great as is the force or least resistance which suffices
to sustain it ;" and the impetus or momentum, in the
course of the proof, being taken to be as the velocity
produced in a given time, it is manifest that the prin
ciple so stated amounts to this ; that the velocity pro
duced is as the statical force. And thus this law of
motion appears, in the school of Galileo, to have been
suggested and established at first by experiment, but
afterwards confirmed and demonstrated by a priori
considerations.
We see, in the above reasoning, a number of abstract
terms introduced which are not, at first at least, very
distinctly defined, as impetus, momentum, &c. Of
these, momentum has been selected, to express that
quantity which, in a moving body, measures the statical
force impressed upon the body. This quantity is, as we
have just seen, proportional to the velocity in a given
body. It is also, in different bodies, proportional to the
mass of the body. This part of the third law of motion
follows from our conception of matter in general as con
sisting of parts capable of addition. A double pressure
must be required to produce the same velocity in a
double mass ; for if the mass be halved, each half will
require an equal pressure ; and the addition, both of the
pressures and of the masses, will take place without dis
turbing the effects.
The measure of the quantity of matter of a body con
sidered as affecting the velocity which pressure produces
in the body, is termed its inertia, as we have already
stated, (p. 190.) Inertia is the property by which a
* Diafago, p. 104.
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 235
large mass of matter requires a greater force than a
small mass, to give it an equal velocity. It belongs to
each portion of matter; and portions of inertia are
added whenever portions of matter are added. Hence
inertia is as the quantity of matter ; which is only an
other way of expressing this third law of motion, so far
as quantity of matter is concerned.
But how do we know the quantity of matter of a
body ? We may reply, that we take the weight as the
measure of the quantity of matter : but we may then be
again asked, how it appears that the weight is propor
tional to the inertia ; which it must be, in order that the
quantity of matter may be proportional to both one and
the other. We answer, that this appears to be true
experimentally, because all bodies fall with equal veloci
ties by gravity, when the known causes of difference are
removed. The observations of falling bodies, indeed,
are not susceptible of much exactness : but experiments
leading to the same result, and capable of great precision,
were made upon pendulums by Newton ; as he relates in
his Prinripia, Book m., prop. 6. They all agreed, he
says, with perfect accuracy : and thus the weight and the
inertia are proportional in all cases, and therefore each
proportional to the quantity of matter as measured by
the other.
The conception of inertia, as we have already seen in
chapter v., involves the notion of action and reaction ;
and thus the laws which involve inertia depend upon the
idea of mutual causation. The rule, that the velocity is
as the force, depends upon the principle of causation,
that the effect is proportional to the cause ; the effect
bring here so estimated as to be consistent both with
the other laws of motion and with experiment.
But here, as in other cases, the question occurs
Is experiment really requisite for the proof of
236 PHILOSOPHY OF THE MECHANICAL SCIENCES.
this law ? If we look to authorities, we shall be not a
little embarrassed to decide. D'Alembert is against the
necessity of experimental proof. " Why," says he *,
" should we have recourse to this principle employed, at
the present day, by everybody, that the force is propor
tional to the velocity? ... a principle resting solely
upon this vague and obscure axiom, that the effect is
proportional to the cause. We shall not examine here,"
he adds, " if this principle is necessarily true ; we shall
only avow that the proofs which have hitherto been
adduced do not appear to us unexceptionable : nor shall
we, with some geometers, adopt it as a purely contingent
truth ; which would be to ruin the certainty of me
chanics, and to reduce it to be nothing more than an
experimental science. We shall content ourselves with
observing," he proceeds, " that certain or doubtful, clear
or obscure, it is useless in mechanics, and consequently
ought to be banished from the science." Though
D'Alembert rejects the third law of motion in this form,
he accepts one of equivalent import, which appears to
him to possess axiomatic certainty ; and this procedure
is in consistence with the course which he takes, of
claiming for the science of mechanics more than mere
experimental truth. On the contrary, Laplace considers
this third law as established by experiment. " Is the
force," he saysf, "proportioned to the velocity? This,"
he replies, " we cannot know a priori, seeing that we
are in ignorance of the nature of moving force : we must
therefore, for this purpose, recur to experience ; for all
which is not a necessary consequence of the few data we
have respecting the nature of things, is, for us, only a re
sult of observation." And again he saysj, "Here, then,
we have two laws of motion, — the law of inertia [the first
law of motion], and the law of the force proportional to
* Di/na»iique, Fref. p. x. t Mec Cel. p. 15. |. P. 18.
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. '237
the velocity, — which are given by observation. They
are the most natural and the most simple laws which we
can imagine, and without doubt they flow from the very
nature of matter ; but this nature being unknown, they
are, for us, only observed facts : the only ones, however,
which mechanics borrows from experience."
It will appear, I think, from the views given in this
and several other parts of the present work, that we can
not with justice say that we have very "few data respect
ing the nature of things," in speculating concerning the
laws of the universe ; since all the consequences which
flow from the relations of our fundamental ideas, neces
sarily regulate our knowledge of things, so far as we
have any such knowledge. Nor can we say that the na
ture of matter is unknown to us, in any sense in which
we can conceive knowledge as possible. The nature ol
matter is no more unknown than the nature of space or
of number. In our conception of matter, as of space
and of number, are involved certain relations, which are
the necessary groundwork of our knowledge ; and any
thing which is independent of these relations, is not un
known, but inconceivable.
It must be already clear to the reader, from the
phraseology employed by these two eminent mathema
ticians, that the question respecting the formation of the
third law of motion can only be solved by a careful con
sideration of what we mean by observation and experi
ence, nature and matter. But it will probably be gene
rally allowed, that, taking into account the explanations
already offered of the necessary conditions of experience
and of the conception of inertia, this law of motion, that
the inertia is as the quantity of matter, is almost or alto
gether self-evident.
C. Action and Reaction are Equal in Moving Bodies.
—When we have to consider bodies as acting upon one
238 PHILOSOPHY OF THE MECHANICAL SCIENCES.
another, and influencing each other's motions, the third
law of motion is still applied ; but along with this, we
also employ the general principle that action and reaction
are equal and opposite. Action and reaction are here to
be understood as momentum produced and destroyed,
according to the measure of action established by the
Third Law of Motion : and the cases in which this prin
ciple is thus employed form so large a portion of those
in which the third law of motion is used, that some
writers (Newton at the head of them) have stated the
equality of action and reaction as the third law of motion.
The third law of motion being once established, the
equality of action and reaction, in the sense of mo
mentum gained and lost, necessarily follows. Thus, if
a weight hanging by a string over the edge of a smooth
level table draw another weight along the table, the
hanging weight moves more slowly than it would do if
not so connected, and thus loses velocity by the con
nexion ; while the other weight gains by the connexion
all the velocity which it has, for if left to itself it would
rest. And the pressures which restrain the descent of the
first bodv and accelerate the motion of the second, are
•/
equal at all instants of time, for each of these pressures
is the tension of the string: and hence, by the third law
of motion, the momentum gained by the one body, and
the momentum lost by the other in virtue of the action
of this string, are equal. And similar reasoning may be
employed in any other case where bodies are connected.
The case where one body does not push or draw,
but strikes another, appeared at first to mechanical rea-
soners to be of a different nature from the others ; but a
little consideration was sufficient to show that a blow
is, in fact, only a short and violent pressure ; and that,
therefore, the general rule of the equality of momentum
lost and gained applies to this as well as to the other cases.
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 239
Thus, in order to determine the case of the direct
action of bodies upon one another, we require no new
Ia\v of motion. The equality of action and reaction,
which enters necessarily into every conception of me
chanical operation, combined with the measure of action
as given by the third law of motion, enables us to trace
the consequences of every case, whether of pressure or
of impact.
7. D 'Alembert 's Principle. — But what will be the
result when bodies do not act directly upon each other,
but are indirectly connected in any way by levers, strings,
pulleys, or in any other manner, so that one part of the
system has a mechanical advantage over another? The
result must still be determined by the principle that
action and reaction balance each other. The action and
reaction, being pressures in one sense, must balance each
other by the laws of statics, for these laws determine
the equilibrium of pressure. Now action and reaction,
according to their measures in the Third Law of Motion,
are momentum gained and lost, when the action is di
rect ; and except the indirect action introduce some
modification of the law, they must have the same mea
sure still. But, in fact, we cannot well conceive any
modification of the law to take place in this case; for
direct action is only one (the ultimate) case of indirect
action. Thus if two heavy bodies act at different points
of a lever, the action of each on the other is indirect ;
but if the two points come together, the action becomes
direct. Hence the rule must be that which we have
already stated ; for if the rule were false for indirect
action, it would also be false for direct action, for which
case we have shown it to be true. And thus we obtain
the general principle, that in any system of bodies which
act on each other, action and reaction, estimated by mo
mentum gained and lost, balance each other according
240 PHILOSOPHY OF THE MECHANICAL SCIENCES.
to the laws of equilibrium. This principle, which is so
general as to supply a key to the solution of all pos
sible mechanical problems, is commonly called UAlem-
lerfs Principle. The experimental proofs which con
vinced men of the truth of the Third Law of Motion
were, many or most of them, proofs of the law in this
extended sense. And thus the proof of D'Alembert's
Principle, both from the idea of mechanical action and
from experience, is included in the proof of the law
already stated.
8. Connexion of Dynamical and Statical Principles.
—The principle of equilibrium of D'Alembert just stated,
is the law which he would substitute for the Third Law
of Motion ; and he would thus remove the necessity for
an independent proof of that law. In like manner, the
Second Law of Motion is by some writers derived from
the principle of the composition of statical forces ; and
they would thus supersede the necessity of a reference to
experiment in that case. Laplace takes this course, and
thus, as we have seen, rests only the First and Third Law
of Motion upon experience. Newton, on the other hand,
recognizes the same connexion of propositions, but for
a different purpose ; for he derives the composition of
statical forces from the Second Law of Motion.
The close connexion of these three principles, the
composition of (statical) forces, the composition of (ac
celerating) forces with velocities, and the measure of
(moving) forces by velocities, cannot be denied ; yet it
appears to be by no means easy to supersede the neces
sity of independent proofs of the two last of these prin
ciples. Both may be proved or illustrated by expe
riment : and the experiments which prove the one are
different from those which establish the other. For
example, it appears by easy calculations, that when we
apply our principles to the oscillations of a pendulum,
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 241
the Second Law is proved by the fact, that the oscilla
tions take place at the same rate in an east and west,
and in a north and south direction : under the same cir
cumstances, the Third Law is proved by our finding that
the time of a small oscillation is proportional to the
square root of the length of a pendulum ; and similar
differences might be pointed out in other experiments,
as to their bearing upon the one law or the other.
9. Mechanical Principles become gradually more
simple and more evident. — I will again point out in
general two circumstances which I have already noticed
in particular cases of the laws of motion. — Truths are
often at first assumed in a form which is far from being
the most obvious or simple ; — and truths once discovered
are gradually simplified, so as to assume the appearance
of self-evident truths.
The former circumstance is exemplified in several of
the instances which we have had to consider. The
assumption that a perpetual motion is impossible pre
ceded the knowledge of the first law of motion. The
assumed equality of the velocities acquired down two in
clined planes of the same height, was afterwards reduced
to the third law of motion by Galileo himself. In the
History *, we have noted Huyghens's assumption of the
equality of the actual descent and potential ascent of the
center of gravity : this was afterwards reduced by Her
man and the Bernoullis, to the statical equivalence of the
solicitations of gravity and the vicarious solicitations of
the effective forces which act on each point ; and finally
to the principle of D'Alembert, which asserts that the
motions gained and lost balance each other.
This assertion of principles which now appear neither
obvious nor self-evident, is not to be considered as a
groundless assumption on the part of the discoverers by
* B. vi. c. v. sect. 2.
VOL. I. W. P. R
242 PHILOSOPHY OF THE MECHANICAL SCIENCES.
whom it was made. On the contrary, it is evidence of
the deep sagacity and clear thought which were requisite
in order to make such discoveries. For these results are
really rigorous consequences of the laws of motion in
their simplest form : and the evidence of them was pro
bably present, though undeveloped, in the minds of the
discoverers. We are told of geometrical students, who,
by a peculiar aptitude of mind, perceived the evidence of
some of the more advanced propositions of geometry
without going through the introductory steps. We must
suppose a similar aptitude for mechanical reasonings,
which, existing in the minds of Stevinus, Galileo, New
ton, and Huyghens, led them to make those assumptions
which finally resolved themselves into the laws of motion.
We may observe further, that the simplicity and evi
dence which the laws of mechanics have at length as
sumed, are much favoured by the usage of words among
the best writers on such subjects. Terms which origi
nally, and before the laws of motion were fully known,
were used in a very vague and fluctuating sense, were
afterwards limited and rendered precise, so that asser
tions which at first appear identical propositions become
distinct and important principles. Thus force, motion,
momentum, are terms which were employed, though in a
loose manner, from the very outset of mechanical specu
lation. And so long as these words retained the vagueness
of common language, it would have been a useless and
barren truism to say that " the momentum is proportional
to the force," or that "a body loses as much motion as
it communicates to another." But when " momentum "
and "quantity of motion" are defined to mean the pro
duct of mass and velocity, these two propositions imme
diately become distinct statements of the third law of
motion and its consequences. In like manner, the asser
tion that "gravity is a uniform force" wras assented to,
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 243
before it was settled what a uniform force was ; but this
assertion only became significant and useful when that
point had been properly determined. The statement
that "when different motions are communicated to the
same body their effects are compounded," becomes the
second law of motion, when we define what composition
of motions is. And the same process may be observed
in other cases.
And thus we see how well the form which science
ultimately assumes is adapted to simplify knowledge.
The definitions which are adopted, and the terms which
become current in precise senses, produce a complete
harmony between the matter and the form of our know
ledge; so that truths which were at first unexpected and
recondite, became familiar phrases, and after a few gene
rations sound, even to common ears, like identical pro
positions.
10. Controversy of the Measure of Force. — In the
History of Mechanics'*"", we have given an account of the
controversy which, for some time, occupied the mathema
ticians of Europe, whether the forces of bodies in motion
should be reckoned proportional to the velocity, or to the
square of the velocity. We need not here recall the
events of this dispute ; but we may remark, that its his
tory, as a metaphysical controversy, is remarkable in this
respect, that it has been finally and completely settled;
for it is now agreed among mathematicians that both
sides were right, and that the results of mechanical action
may be expressed with equal correctness by means of
momentum and of vis viva. It is, in one sense, as D'Alem-
bert has saidf, a dispute about words; but we are not
* B. vi. c. v. sect. 2.
t D'Alcinbcrt has also remarked (Dynatniqtie, Pref. xxi.,) that this
controversy " shows how little justice and precision there is in the
pretended axiom that causes are proportional to their effects." But
R2
244 PHILOSOPHY OF THE MECHANICAL SCIENCES.
to infer that, on that account, it was frivolous or useless ;
for such disputes are one principal means of reducing the
principles of our knowledge to their utmost simplicity
and clearness. The terms which are employed in the
science of mechanics are now liberated for ever, in the
minds of mathematicians, from that ambiguity which
was the battle-ground in the war of the vis viva.
But we may observe that the real reason of this con
troversy was exactly that tendency which we have been
noticing ; — the disposition of man to assume in his specu
lations certain general propositions as true, and to fix the
sense of terms so that they shall fall in with this truth.
It was agreed, on all hands, that in the mutual action of
bodies the same quantity of force is always preserved;
and the question was, by which of the two measures this
rule could best be verified. We see, therefore, that the
dispute was not concerning a definition merely, but con
cerning a definition combined with a general proposition.
Such a question may be readily conceived to have been
by no means unimportant ; and we may remark, in pass
ing, that such controversies, although they are commonly
afterwards stigmatized as quarrels about words and defi
nitions, are, in reality, events of considerable conse
quence in the history of science ; since they dissipate all
ambiguity and vagueness in the use of terms, and bring
into view the conditions under which the fundamental
principles of our knowledge can be most clearly and
simply presented.
It is worth our while to pause for a moment on the
prospect that we have thus obtained, of the advance of
this reflection is by no means well founded. For since both measures
are true, it appears that causes may be justly measured by their effects,
even when very different kinds of effects are taken. That the axiom
does not point out one precise measure, till illustrated by experience or j
by other considerations, we grant : but the same thing occurs in the
application of other axioms also.
ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 245
knowledge, as exemplified in the history of Mechanics.
The general transformation of our views from vague to
definite, from complex to simple, from unexpected dis
coveries to self-evident truths, from seeming contradic
tions to identical propositions, is very remarkable, but it
is by no means peculiar to our subject. The same cir
cumstances, more or less prominent, more or less deve
loped, appear in the history of other sciences, according
to the point of advance which each has reached. They
bear upon very important doctrines respecting the pro
spects, the limits, and the very nature of our knowledge.
And though these doctrines require to be considered with
reference to the whole body of science, yet the peculiar
manner in which they are illustrated by the survey of
the history of Mechanics, on which we have just been
engaged, appears to make this a convenient place for
introducing them to the reader.
CHAPTER VIII.
OF THE PARADOX OF UNIVERSAL PROPOSI
TIONS OBTAINED FROM EXPERIENCE.
1. IT was formerly stated"- that experience cannot
establish any universal or necessary truths. The number
of trials which we can make of any proposition is neces
sarily limited, and observation alone cannot give us any
ground of extending the inference to untried cases. Ob
served facts have no visible bond of necessary connexion,
and no exercise of our senses can enable us to discover
such connexion. We can never acquire from a mere
observation of facts, the right to assert that a proposition
is true in all cases, and that it could not be otherwise
than we find it to be.
* B. i., c. v. Of Experience.
246 PHILOSOPHY OF THE MECHANICAL SCIENCES.
Yet, as we have just seen in the history of the laws of
motion, we may go on collecting our knowledge from
observation, and enlarging and simplifying it, till it ap
proaches or attains to complete universality and seeming
necessity. Whether the laws of motion, as we now know
them, can be rigorously traced to an absolute necessity in
the nature of things, we have not ventured absolutely to
pronounce. But we have seen that some of the most
acute and profound mathematicians have believed that,
for these laws of motion, or some of them, there was
such a demonstrable necessity compelling them to be
such as they are, and no other. Most of those who have
carefully studied the principles of Mechanics will allow
that some at least of the primary laws of motion approach
very near to this character of necessary truth ; and will
confess that it would be difficult to imagine any other
consistent scheme of fundamental principles. And almost
all mathematicians will allow to these lawrs an absolute
universality ; so that we may apply them without scruple
or misgiving, in cases the most remote from those to
which our experience has extended. What astronomer
would fear to refer to the known laws of motion, in rea
soning concerning the double stars; although these objects
are at an immeasurably remote distance from that solar
system which has been the only field of our observation
of mechanical facts? What philosopher, in speculating
respecting a magnetic fluid, or a luminiferous ether, would
hesitate to apply to it the mechanical principles which
are applicable to fluids of known mechanical properties ?
When we assert that the quantity of motion in the world
cannot be increased or diminished by the mutual actions
of bodies, does not every mathematician feel convinced
that it would be an unphilosophical restriction to limit
this proposition to such modes of action as we have
tried?
PARADOX OF UNIVERSAL PROPOSITIONS. 247
Yet no one can doubt that, in historical fact, these
laws were collected from experience. That such is the
case, is no matter of conjecture. We know the time, the
persons, the circumstances, belonging to each step of each
discovery. I have, in the History, given an account of
these discoveries ; and in the previous chapters of the pre
sent work, I have further examined the nature and the
import of the principles which were thus brought to light.
Here, then, is an apparent contradiction. Experi
ence, it would seem, has done that which we had proved
that she cannot do. She has led men to propositions,
universal at least, and to principles which appear to some
persons necessary. What is the explanation of this con
tradiction, the solution of this paradox ? Is it true that
Experience can reveal to us universal and necessary
truths ? Does she possess some secret virtue, some un
suspected power, by which she can detect connexions
and consequences which we have declared to be out of
her sphere? Can she see more than mere appearances,
and observe more than mere facts ? Can she penetrate,
in some way, to the nature of things ? — descend below the
surface of phenomena to their causes and origins, so as
to be able to say what can and what can not be ;• — what
occurrences are partial, and what universal ? If this be
so, we have indeed mistaken her character and powers ;
and the whole course of our reasoning becomes pre
carious and obscure. But, then, when we return upon
our path we cannot find the point at which we deviated,
we cannot detect the false step in our deduction. It
still seems that by experience, strictly so called, we
cannot discover necessary and universal truths. Our
senses can give us no evidence of a necessary connexion
in phenomena. Our observation must be limited, and
cannot testify concerning anything which is beyond its
limits. A general view of our faculties appears to prove
248 PHILOSOPHY OF THE MECHANICAL SCIENCES.
it to be impossible that men should do what the history
of the science of mechanics shows that they have done.
2. But in order to try to solve this Paradox, let us
again refer to the History of Mechanics. In the cases
belonging to that science, in which propositions of the
most unquestionable universality, and most approaching
to the character of necessary truths, (as, for instance, the
laws of motion,) have been arrived at, what is the source
of the axiomatic character which the propositions thus
assume ? The answer to this question will, we may hope,
throw some light on the perplexity in which we appear
to be involved.
Now the answer to this inquiry is, that the laws
of motion borrow their axiomatic character from their
being merely interpretations of the Axioms of Causation.
Those axioms, being exhibitions of the Idea of Cause
under various aspects, are of the most rigorous univer
sality and necessity. And so far as the laws of motion
are exemplifications of those axioms, these laws must be
no less universal and necessary. How these axioms are
to be understood ; — in what sense cause and effect, action
and reaction, are to be taken, experience and observa
tion did, in fact, teach inquirers on this subject ; and
without this teaching, the laws of motion could never
have been distinctly known. If two forces act together,
each must produce its effect, by the axiom of causation ;
and, therefore, the effects of the separate forces must be
confounded. But a long course of discussion and expe
riment must instruct men of what kind this composition
of forces is. Again ; action and reaction must be equal ;
but much thought and some trial were needed to show
what action and reaction are. Those metaphysicians who
enunciated Laws of motion without reference to expe
rience, propounded only such laws as were vague and
inapplicable. But yet these persons manifested the
PARADOX OF UNIVERSAL PROPOSITIONS. 249
indestructible conviction, belonging to man's speculative
nature, that there exist Laws of motion, that is, uni
versal formula?, connecting the causes and effects when
motion takes place. Those mechanicians, again, who,
observed facts involving equilibrium and motion, and
stated some narrow rules, without attempting to ascend
to any universal and simple principle, obtained laws no
less barren and useless than the metaphysicians ; for
they could not tell in what new cases, or whether in
any, their laws would be verified ; — they needed a more
general rule, to show them the limits of the rule they
had discovered. They went wrong in each attempt to
solve a new problem, because their interpretation of
the terms of the axioms, though true, perhaps, in certain
cases, was not right in general.
Thus Pappus erred in attempting to interpret as a
case of the lever, the problem of supporting a weight
upon an inclined plane ; thus Aristotle erred in inter
preting the doctrine that the weight of bodies is the
cause of their fall ; thus Kepler erred in interpreting the
rule that the velocity of bodies depends upon the force;
thus Bernoulli'-" erred in interpreting the equality of
action and reaction upon a lever in motion. In each
of these instances, true doctrines, already established,
(whether by experiment or otherwise,) were erroneously
applied. And the error was corrected by further reflec
tion, which pointed out that another mode of interpreta
tion was requisite, in order that the axiom which was
appealed to in each case might retain its force in the
most general sense. And in the reasonings which avoided
or corrected such errors, and which led to substantial
general truths, the object of the speculator always was
to give to the acknowledged maxims which the Idea of
Cause suggested, such a signification as should be con-
* Hist. Ind. .VcJ., B. vi. c. v. st-ct. 2.
'250 PHILOSOPHY OF THE MECHANICAL SCIENCES.
sistcnt with their universal validity. The rule was not
accepted as particular at the outset, and afterwards gene
ralized more and more widely ; but from the very first,
the universality of the rule was assumed, and the ques
tion was, how it should be understood so as to be
universally true. At every stage of speculation, the law
was regarded as a general law. This was not an aspect
which it gradually acquired, by the accumulating con
tributions of experience, but a feature of its original
and native character. What should happen universally,
experience might be needed to show : but that what
happened should happen universally, was implied in the
nature of knowledge. The universality of the laws of
motion was not gathered from experience, however much
the laws themselves might be so.
3. Thus we obtain the solution of our Paradox, so
far as the case before us is concerned. The laws of
motion borrow their form from the Idea of Causation,
though their matter may be given by experience : and
hence they possess a universality which experience cannot
give. They are certainly and universally valid ; and the
only question for observation to decide is, how they are
to be understood. They are like general mathematical
formula?, which are known to be true, even while we are
ignorant what are the unknown quantities which they
involve. It must be allowed, on the other hand, that so
long as these formula? are not interpreted by a real
study of nature, they are not only useless but prejudi
cial ; filling men's minds with vague general terms, empty
maxims, and unintelligible abstractions, which they mis
take for knowledge. Of such perversion of the specula
tive propensities of man's nature, the world has seen too
much in all ages. Yet we must not, on that account,
despise these forms of truth, since without them, no
general knowledge is possible. Without general terms,
PARADOX OF UNIVERSAL PROPOSITIONS. 251
and maxims, and abstractions, we can have no science,
no speculation ; hardly, indeed, consistent thought or
the exercise of reason. The course of real knowledge is,
to obtain from thought and experience the right inter
pretation of our general terms, the real import of our
maxims, the true generalizations which our abstractions
involve.
4. If it be asked, How Experience is able to teach us
to interpret aright the general terms which the Axioms
of Causation involve ; — whence she derives the light
which she is to throw on these general notions ; the
answer is obvious ; — namely, that the relations of causa
tion are the conditions of Experience; — that the general
notions are exemplified in the particular cases of which
she takes cognizance. The events which take place
about us, and which are the objects of our observation,
we cannot conceive otherwise than as subject to the
laws of cause and effect. Every event must have a
cause; — Every effect must be determined by its cause; —
these maxims are true of the phenomena which form
the materials of our experience. It is precisely to them,
that these truths apply. It is in the world which we
have before our eyes, that these propositions are univer
sally verified ; and it is therefore by the observation of
what we see, that we must learn how these propositions
are to be understood. Every fact, every experiment, is
an example of these statements ; and it is therefore by
attention to and familiarity with facts and experiments,
that we learn the signification of the expressions in which
the statements are made ; just as in any other case we
learn the import of language by observing the manner
in which it is applied in known cases. Experience is
the interpreter of nature ; it being understood that she
is to make her interpretation in that comprehensive
phraseology which is the genuine language of science.
'2o'2 PHILOSOPHY OF THE MECHANICAL SCIENCES.
5. We may return for an instant to the objection,
that experience cannot give us general truths, since,
after any number of trials confirming a rule, we may,
for aught we can foresee, have one which violates the
rule. When we have seen a thousand stones fall to the
ground, we may see one which does not fall under the
same apparent circumstances. How then, it is asked,
can experience teach us that all stones, rigorously speak
ing, will fall if unsupported ? And to this we reply,
that it is not true that we can conceive one stone to be
suspended in the air, while a thousand others fall, with
out believing some peculiar cause to support it ; and
that, therefore, such a supposition forms no exception to
the law, that gravity is a force by which all bodies are
urged downwards. Undoubtedly we can conceive a body,
when dropt or thrown, to move in a line quite different
from other bodies: thus a certain missile* used by the
natives of Australia, and lately brought to this country,
when thrown from the hand in a proper manner, de
scribes a curve, and returns to the place from whence it
was thrown. But did any one, therefore, even for an
instant suppose that the laws of motion are different for
this and for other bodies? On the contrary, was not
every person of a speculative turn immediately led to
inquire how it was that the known causes which modify
motion, the resistance of the air and the other causes,
produced in this instance so peculiar an effect ? And if
the motion had been still more unaccountable, it would
not have occasioned any uncertainty whether it were
consistent with the agency of gravity and the laws of
motion. If a body suddenly alter its direction, or move
in any other unexpected manner, we never doubt that
there is a cause of the change. We may continue quite
ignorant of the nature of this cause, but this ignorance
* Called the Bo-me-rang.
PARADOX OF UNIVERSAL PROPOSITIONS. 253
never occasions a moment's doubt that the cause exists
and is exactly suited to the effect. And thus experience
can prove or discover to us general rules, but she can
never prove that general rules do not exist. Anomalies,
exceptions, unexplained phenomena, may remind us that
we have much still to learn, but they can never make
us suppose that truths are not universal. We may ob
serve facts that show us we have not fully understood
the meaning of our general laws, but we can never find
facts which show our laws to have no meaning. Our
experience is bound in by the limits of cause and effect,
and can give us no information concerning any region
where that relation does not prevail. The whole series
of external occurrences and objects, through all time
and space, exists only, and is conceived only, as subject
to this relation ; and therefore we endeavour in vain to
imagine to ourselves when and where and how excep
tions to this relation may occur. The assumption of the
connexion of cause and effect is essential to our expe
rience, as the recognition of the maxims which express
this connexion is essential to our knowledge.
6. I have thus endeavoured to explain in some
measure how, at least in the field of our mechanical
knowledge, experience can discover universal truths,
though she cannot give them their universality ; and
how such truths, though borrowing their form from our
ideas, cannot be understood except by the actual study
of external nature. And thus with regard to the laws
of motion, and other fundamental principles of Mechanics,
the analysis of our ideas and the history of the progress
of the science well illustrate each other.
If the paradox of the discovery of universal truths
by experience be thus solved in one instance, a much
wider question offers itself to us ; — How far the difficulty,
and how far the solution, are applicable to other sub-
•J54 PHILOSOPHY OF THE MECHANICAL SCIENCES.
jects. It is easy to see that this question involves most
grave and extensive doctrines with regard to the whole
compass of human knowledge : and the views to which
we have been led in the present Book of this work are,
we trust, fitted to throw much light upon the general
aspect of the subject. But after discussions so abstract,
and perhaps obscure, as those in which we have been
engaged for some chapters, I willingly postpone to a
future occasion an investigation which may perhaps
appear to most readers more recondite and difficult
still. And we have, in fact, many other special fields
of knowledge to survey, before we are led by the order
of our subject, to those general questions and doctrines,
those antitheses brought into view and again resolved,
which a view of the whole territory of human know
ledge suggests, and by which the nature and conditions
of knowledge are exhibited.
Before we quit the subject of mechanical science we
shall make a few remarks on another doctrine which
forms part of the established truths of the science,
namely, the doctrine of universal gravitation.
CHAPTER IX.
OF THE ESTABLISHMENT OF THE LAW OF
UNIVERSAL GRAVITATION.
THE doctrine of universal gravitation is a feature of so
much importance in the history of science that we shall
not pass it by without a few remarks on the nature and
evidence of the doctrine.
1. To a certain extent the doctrine of the attraction
of bodies according to the law of the inverse square of
the distance, exhibits in its progress among men the
ESTABLISHMENT OF UNIVERSAL GRAVITATION. 255
same general features which we have noticed in the his
tory of the laws of motion. This doctrine was main
tained a priori on the ground of its simplicity, and as
serted positively, even before it was clearly understood :
— notwithstanding this anticipation, its establishment
on the ground of facts was a task of vast labour and
sagacity : — when it had been so established in a general
way, there occurred at later periods, an occasional sus
picion that it might be approximately true only : — these
suspicions led to further researches, which showed the
rule to be rigorously exact : — and at present there are
mathematicians who maintain, not only that it is true,
but that it is a necessary property of matter. A very
few words on each of these points will suffice.
2. I have shown in the History of Science*, that the
attraction of the sun according to the inverse square of
the distance, had been divined by Bullialdus, Hooke,
Halley, and others, before it was proved by Newton.
Probably the reason which suggested this conjecture was,
that gravity might be considered as a sort of emanation ;
and that thus, like light or any other effect diffused from
a center, it must follow the law just stated, the efficacy
of the force being weakened in receding from the center,
exactly in proportion to the space through which it is
diffused. It cannot be denied that such a view appears
to be strongly recommended by analogy.
When it had been proved by Newton that the planets
were really retained in thein elliptical orbits by a central
force, his calculations also showed that the above-stated
law of the force must be at least very approximately
correct, since otherwise the aphelia of the orbits could
not be so nearly at rest as they were. Yet when it
seemed as if the motion of the moon's apogee could not
be accounted for without some new supposition, the d
* B. VTI. c. i.
256 PHILOSOPHY OF THE MECHANICAL SCIENCES.
priori argument in favour of the inverse square did not
prevent Clairaut from trying the hypothesis of a small
term added to that which expressed the ancient law :
but when, in order to test the accuracy of this hypothe
sis, the calculation of the motion of the moon's apogee
was pushed to a greater degree of exactness than had
been obtained before, it was found that the new term
vanished of itself; and that the inverse square now ac
counted for the whole of the motion. And thus, as in
the case of the second law of motion, the most scrupulous
examination terminated in showing the simplest rule to
be rigorously true.
3. Similar events occurred in the history of another
part of the law of gravitation : namely, that the attrac
tion is proportional to the quantity of matter attracted.
This part of the law may also be thus stated, That the
weight of bodies arising from gravity is proportional to
their inertia ; and thus, that the accelerating force on
all bodies under the same circumstances is the same.
Newton made experiments which proved this with re
gard to terrestrial bodies ; for he found that, at the end
of equal strings, balls of all substances, gold, silver,
lead, glass, wood, &c., oscillated in equal times'". But
a few years ago, doubts arose among the German astro
nomers whether this law was rigorously true with regard
to the planetary bodies. Some calculations appeared
to prove, that the attraction of Jupiter as shown by the
perturbations which he produces in the small planets
Juno, Vesta, and Pallas, was different from the attrac
tion which he exerts on his own satellites. Nor did
there appear to these philosophers anything inconceiv
able in the supposition that the attraction of a planet
might be thus elective. But when Mr. Airy obtained
a more exact determination of the mass of Jupiter, as
* Prin. Lib. in., Prop. (>.
ESTABLISHMENT OF UNIVERSAL GRAVITATION. 257
indicated by his effect on his satellites, it was found
that this suspicion was unfounded ; and that there was,
in this case, no exception to the universality of the rule,
that this cosmical attraction is in the proportion of the
attracted mass.
4. Again : when it had thus been shown that a
mutual attraction of parts, according to the law above
mentioned, prevailed throughout the extent of the solar
system, it might still be doubted whether the same law
extended to other regions of the universe. It might
have been perhaps imagined that each fixed star had
its peculiar law of force. But the examination of the
motions of double stars about each other, by the two
Ilerschels and others, appears to show that these bodies
describe ellipses as the planets do ; and thus extends the
law of the inverse squares to parts of the universe im
measurably distant from the whole solar system.
5. Since every doubt which has been raised with
regard to the universality and accuracy of the law of
gravitation, has thus ended in confirming the rule, it is
not surprizing that men's minds should have returned
with additional force to those views which had at first
represented the law as a necessary truth, capable of being
established by reason alone. When it had been proved
by Newton that gravity is really a universal attribute of
matter as far as we can learn, his pupils were not con
tent without maintaining it to be an essential quality.
This is the doctrine held by Cotes in the preface to the
second edition of the Principia (1712): "Gravity," he
says, " is a primary quality of bodies, as extension, mo
bility, and impenetrability are." But Newton himself
by no means went so far. In his second Letter to
Bentley (1G03), he says: "You sometimes speak of
gravity as essential and inherent to matter; pray do
not ascribe that notion to me. The cause of gravity,"
VOL. i. w. P. S
258 PHILOSOPHY OF THE .MECHANICAL SCIENCES.
he adds, " I do not pretend to know, and would take
more time to consider of it."
Cotes maintains his opinion by urging, that we learn
by experience that all bodies possess gravity, and that we
do not learn in any other way that they are extended,
movcable, or solid. But we have already seen, that the
ideas of space, time, and reaction, on which depend
extension, mobility, and solidity, are not results, but
conditions, of experience. We cannot conceive a body
except as extended ; we cannot conceive it to exert
mechanical action except with some kind of solidity.
But so far as our conceptions of body have hitherto
been developed, we find no difficulty in conceiving two
bodies which do not attract each other.
G. Newton lays down, in the second edition of the
Princijria, this " Rule of Philosophizing" (Book in.) ;
that " The qualities of bodies which cannot be made
more or less intense, and which belong to all bodies on
which we are able to make experiments, are to be held
to be qualities of all bodies in general." And this Rule
is cited in the sixth Proposition of the Third Book of
the Principia, (Cor. 2,) in order to prove that gravity,
proportional to the quantity of matter, may be asserted
to be a quality of all bodies universally. But we may
remark that a Rule of Philosophizing, itself of precarious
authority, cannot authorize us in ascribing universality
to an empirical result. Geometrical and statical pro
perties are seen to be necessary, and therefore universal :
but Newton appears disposed to assert a like universality
of gravity, quite unconnected with any necessity. It
would be a very inadequate statement, indeed a false
representation, of statical truth, if we were to say, that
because every body which has hitherto been tried has
been found to have a center of gravity, we venture to
assert that all bodies whatever have a center of gravity.
ESTABLISHMENT OF UNIVERSAL GRAVITATION. 250
And if we arc ever able to assert the absolute univer
sality of the law of gravitation, we shall have to rest
this truth upon the clearer developement of our ideas of
matter and force ; not upon a Rule of Philosophizing,
which, till otherwise proved, must be a mere rule of
prudence, and which the opponent may refuse to admit.
7. Other persons, instead of asserting gravity to be
in its own nature essential to matter, have made hypo
theses concerning some mechanism or other, by which
this mutual attraction of bodies is produced"-. Thus
the Cartesians ascribed to a vortex the tendency of
bodies to a center ; Newton himself seems to have been
disposed to refer this tendency to the elasticity of an
ether; Le Sage propounded a curious hypothesis, in
which this attraction is accounted for by the impulse
of infinite streams of particles flowing constantly through
the universe in all directions. In these speculations,
the force of gravity is resolved into the pressure or im
pulse of solids or fluids. On the other hand, hypotheses
have been propounded, in which the solidity, and other
physical qualities of bodies, have been explained by
representing the bodies as a collection of points, from
which points, repulsive, as well as attractive, forces
emanate. This view of the constitution of bodies was
maintained and developed by Boscovich, and is hence
termed " Boscovich's Theory :" and the discussion of it
will more properly come under our review at a future
period, when we speak of the question whether bodies
are made up of atoms. But we may observe, that New
ton himself appears to have inclined, as his followers
certainly did, to this mode of contemplating the physical
properties of bodies. In his Preface to the Principia,
after speaking of the central forces which are exhibited
* Sec Vince, Observations on the Hyjyothcsis respecting Gravitation,
and the Critique of that work, Edinb. Rev. Vol. xui.
82
•JGO PHILOSOPHY OF THE .MECHANICAL SCIENCES.
in cosmical phenomena, he says : " Would that we could
derive the other phenomena of Nature from mechanical
principles by the same mode of reasoning. For many
things move me, so that I suspect all these phenomena
may depend upon certain forces, by which the particles
of bodies, through causes not yet known, are either im
pelled to each other and cohere according to regular
figures, or are repelled and recede from each other :
which forces being unknown, philosophers have hitherto
made their attempts upon nature in vain."
8. But both these hypotheses ; — that by which cohe
sion and solidity are reduced to attractive and repulsive
forces, and that by which attraction is reduced to the
impulse and pressure of media; — are hitherto merely
modes of representing mechanical laws of nature ; and
cannot, either of them, be asserted as possessing any evi
dent truth or peremptory authority to the exclusion of
the other. This consideration may enable us to estimate
the real weight of the difficulty felt in assenting to the
mutual attraction of bodies not in contact with each
other ; for it is often urged that this attraction of bodies
at a distance is an absurd supposition.
The doctrine is often thus stigmatized, both by popu
lar and by learned writers. It was long received as a
maxim in philosophy (as Monboddo informs us*), that a
body cannot act where it is not, any more than when it
is not. But to this we reply, that time is a necessary
condition of our conception of causation, in a different
manner from space. The action of force can only be
conceived as taking place in a succession of moments, in
each of which cause and effect immediately succeed each
other : and thus the interval of time between a cause and
its remote effect is filled up by a continuous succession
of events connected by the same chain of causation. But
Ancient Metaphysics, Vol. n. p. 17").
ESTADLISIIMENT OF UNIVERSAL GRAVITATION.
in space, there is no such visible necessity of continuity ;
the action and reaction may take place at a distance from
each other ; all that is necessary being that they be
equal and opposite.
Undoubtedly the existence of attraction is rendered
more acceptable to common apprehension by supposing
some intermediate machinery, — a cord, or rod, or fluid,
— by which the forces may be conveyed from one point
to another. But such images are rather fitted to satisfy
those prejudices which arise from the earlier application
of our ideas of force, than to exhibit the real nature of
those ideas. If we suppose two bodies to pull each other
by means of a rod or a cord, we only suppose, in addition
to those equal and opposite forces acting upon the two
bodies which forces are alone essential to mutual attrac
tion, a certain power of resisting transverse pressure at
every point of the intermediate line : which additional
supposition is entirely useless, and quite unconnected
with the essential conditions of the case. When the New
tonians were accused of introducing into philosophy an
unknown cause which they termed attraction, they justly
replied that they knew as much respecting attraction
as their opponents did about impulse. In each case we
have a knowledge of the conception in question so far as
we clearly apprehend it under the conditions of those
axioms of mechanical causation which form the basis of
our science on such subjects.
Having thus examined the degree of certainty and
generality to which our knowledge of the law of univer
sal gravitation has been carried, by the progress of
mechanical discovery and speculation up to the present
time, we might proceed to the other branches of science,
and examine in like manner their grounds and conditions.
But before we do this, it will be worth our while to
attend for a moment to the effect which the progress of
2()2 PHILOSOPHY OF THE MECHANICAL SCIENCES.
mechanical ideas among mathematicians and mechanical
philosophers has produced upon the minds of other per
sons, who share only in an indirect and derivative man
ner in the influence of science.
CHAPTER X.
OF THE GENERAL DIFFUSION OF CLEAR
MECHANICAL IDEAS.
1. WE have seen how the progress of knowledge
upon the subject of motion and force has produced, in
the course of the world's history, a great change in the
minds of acute and speculative men ; so that such per
sons can now reason with perfect steadiness and precision
upon subjects on which, at first, their thoughts were
vague and confused ; and can apprehend, as truths of
complete certainty and evidence, laws which it required
great labour and time to discover. This complete deve-
lopement and clear manifestation of mechanical ideas
has taken place only among mathematicians and philo
sophers. But yet a progress of thought upon such
subjects, — an advance from the obscure to the clear, and
from errour to truth, — may be traced in the world at
large, and among those who have not directly cultivated
the exact sciences. This diffused and collateral influence
of science manifests itself, although in a wavering and
fluctuating manner, by various indications, at various
periods of literary history. The opinions and reasonings
which are put forth upon mechanical subjects, and above
all, the adoption, into common language, of terms and
phrases belonging to the prevalent mechanical systems,
exhibit to us the most profound discoveries and specula
tions of philosophers in their effect upon more common
DIFFUSION OF CLEAR MECHANICAL IDEAS. 263
and familiar trains of thought. This effect is by no
means unimportant, and we shall point out some ex
amples of such indications as we have mentioned.
2. The discoveries of the ancients in speculative
mechanics were, as we have seen, very scanty ; and
hardly extended their influence to the unmathematical
world. Yet the familiar use of the term " center of
of gravity" preserved and suggested the most important
part of what the Greeks had to teach. The other phrases
which they employed, as momentum, energy, virtue,
force, and the like, never had any exact meaning, even
among mathematicians ; and therefore never, in the
ancient world, became the means of suggesting just
habits of thought. I have pointed out, in the History
of Science, several circumstances which appear to denote
the general confusion of ideas which prevailed upon
mechanical subjects during the times of the Roman
empire. I have there taken as one of the examples of
this confusion, the fable narrated by Pliny and others
concerning the echinei's, a small fish, which was said to
stop a ship merely by sticking to it*. This story was
adduced as betraying the absence of any steady appre
hension of the equality of action and reaction ; since the
fish, except it had some immoveable obstacle to hold by,
must be pulled forward by the ship, as much as it pulled
the ship backward. If the writers who speak of this
wonder had shown any perception of the necessity of
a reaction, either produced by the rapid motion of the
fish's fins in the water, or in any other way, they would
not be chargeable with this confusion of thought ; but
from their expressions it is, I think, evident that they
saw no such necessity f. Their idea of mechanical action
* Hist. Ind. Sci. B. iv. c. i. sect. 2.
t Sec Prof. Powell, On the Nature and Evidence of the Lan-s of
Motion. Reports of the Ashmolean Society. Oxford. 1K37- Professor
204 PHILOSOPHY OF THE MKCIIAMC'AL SCIENCES.
was not sufficiently distinct to enable them to sec the
absurdity of supposing an intense pressure with no
obstacle for it to exert itself against.
3. We may trace, in more modern times also, indica
tions of a general ignorance of mechanical truths. Thus
the phrase of shooting at an object "point-blank," im
plies the belief that a cannon-ball describes a path of
which the first portion is a straight line. This error
was corrected by the true mechanical principles which
Galileo and his followers brought to light; but these
principles made their way to popular notice, principally
in consequence of their application to the motions of the
solar system, and to the controversies which took place
respecting those motions. Thus by far the most power
ful argument against the reception of the Copernican
system of the universe, was that of those who asked,
Why a stone dropt from a tower was not left behind by
the motion of the earth ? The answer to this question,
now universally familiar, involves a reference to the true
doctrine of the composition of motions. Again; Kepler's
persevering and strenuous attempts* to frame a phy
sical theory of the universe were frustrated by his igno
rance of the first law of motion, which informs us that
a body will retain its velocity without any maintaining
force. He proceeded upon the supposition that the sun's
force was requisite to keep up the motion of the planets,
Powell has made an objection to my use of this instance of confusion
of thought ; the remark in the text seems to me to justify what 1 said
in the History. As an evidence that the fish was not supposed to pro
duce its effect by its muscular power acting on the water, we may take
what Pliny says, Nat. Hist., xxxii. l,"T)omat mundi rabiem, nullo
siio labore ; non retinendo, aut alio modo quam adluerendo :" and also
wi.at he states in another place (ix. 41,) that when it is preserved in
pickle, it may be used in recovering gold which has fallen into a deep
well. All tins implies adhesion alone, with no conception of reaction.
* 11'*!. I,,'!, ,SW., B. .,. c. iv.. ;,n,l B. vn. c. i.
DIFFUSION OF CLEAR MECHANICAL IDEAS. 2()f>
as well as to deflect and modify it ; and he was thus
led to a system which represented the sun as carrying
round the planets in their orbits by means of a vortex,
produced by his revolution. The same neglect of the
laws of motion presided in the formation of Descartes'
system of vortices. Although Descartes had enunciated
in words the laws of motion, he and his followers showed
that they had not the practical habit of referring to
these mechanical principles ; and dared not trust the
planets to move in free space without some surrounding
machinery to support them*.
4. When at last mathematicians, following Newton,
had ventured to consider the motion of each planet as a
mechanical problem not different in its nature from the
motion of a stone cast from the hand ; and when the
solution of this problem and its immense consequences
had become matters of general notoriety and interest ;
the new views introduced, as is usual, new terms, which
soon became extensively current. We meet with such
phrases as " flying off in the tangent," and " deflexion
from the tangent;" with antitheses between "centripetal"
and "centrifugal force," or between "projectile" and
" central force." " Centers of force," " disturbing forces,"
"perturbations," and "perturbations of higher orders,"
arc not unfrequently spoken of: and the expression "to
gravitate," and the term "universal gravitation," acquired
a permanent place in the language.
Yet for a long time, and even up to the present day,
we find many indications that false and confused appre
hensions on such subjects arc by no means extirpated.
* I have, in the History, applied to Descartes the character which
Iiacon gives to Aristotle, " Audax siimil et paviclus :" though he was
hold enough to enunciate the laws of motion without knowing them
aright, he had not the courage to leave the planets to describe their
orbits by the agency of those laws, without the machinery of contact.
206 PHILOSOPHY OF THE MECHANICAL SCIENCES.
Arguments arc urged against the mechanical system of
the universe, implying in the opponents an absence of
all clear mechanical notions. Many of this class of
writers retrograde to Kepler's point of view. This is,
for example, the case with Lord Monboddo, who, arguing
on the assumption that force is requisite to maintain, as
well as to deflect motion, produced a series of attacks
upon the Newtonian philosophy ; which he inserted in
his Ancient Metaphysics, published in 1771) and the
succeeding years. This writer (like Kepler), measures
force by the velocity which the body has*., not by that
which its gains. Such a use of language would prevent
our obtaining any laws of motion at all. Accordingly,
the author, in the very next page to that which I have
just quoted, abandons this measure of force, and, in cur
vilinear motion, measures force by " the fall from the
extremity of the arc." Again ; in his objections to the
received theory, he denies that curvilinear motion is
compounded, although his own mode of considering such
motion assumes this composition in the only way in
which it was ever intended by mathematicians. Many
more instances might be adduced to show that a want
of cultivation of the mechanical ideas rendered this phi-
, losopher incapable of judging of a mechanical system.
The following extract from the Ancient Metaphy
sics, may be sufficient to show the value of the author's
criticism on the subjects of which we are now speaking.
His object is to prove that there do not exist a centri
petal and a centrifugal force in the case of elliptical
motion. "Let any man move in a circular or elliptical
line described to him ; and he will find no tendency in
himself cither to the center or from it, much less both.
If indeed he attempt to make the motion with great
velocity, or if he do it carelessly and inattentively, he
* Anc. Met. Vol. IT. K v. p. vi.. p. 413.
DIFFUSION OF CLEAR MECHANICAL IDEAS. 207
may go out of the line, either towards the center or from
it : hut this is to be ascribed, not to the nature of the
motion, but to our infirmity ; or perhaps to the animal
form, which is more fitted for progressive motion in a
right line than for any kind of curvilinear motion. But
this is not .the case with a sphere or spheroid, which is
equally adapted to motion in all directions'"." We need
hardly remind the reader that the manner in which a
man running round a small circle, finds it necessary to
lean inwards, in order that there may be a centripetal
inclination to counteract the centrifugal force, is a
standard example of our mechanical doctrines ; and this
fact (quite familiar in practice as well as theory,) is in
direct contradiction of Lord Monboddo's assertion.
5. A similar absence of distinct mechanical thought
appears in some of the most celebrated metaphysicians
of Germany. I have elsewhere noted f the opinion ex
pressed by Hegel, that the glory which belongs to Kepler
has been unjustly transferred to Newton ; and I have
suggested, as the explanation of this mode of thinking,
that Hegel himself, in the knowledge of mechanical
truth, had not advanced beyond Kepler's point of view.
Persons who possess conceptions of space and number,
but who have not learnt to deal with ideas of force and
causation, may see more value in the discoveries of Kepler
than in those of Newton. Another exemplification of
this state of mind may be found in Mr. Schelling's spe
culations ; for instance, in his Lectures on the Method of
Academical Study. In the twelfth Lecture, on the Study
of Physics and Chemistry, he says, (p. 26G,) " What the
mathematical natural philosophy has done for the know
ledge of the laws of the universe since the time that
they were discovered by his (Kepler's) godlike genius, is,
* Anc. Met., Vol. i. R ii. c. 10, p. 2(51.
t Hisl. Ind. Sci.. B. vn. c. ii. sect. />.
*J()8 PHILOSOPHY OF THE MECHANICAL SCIENCES.
as is well known, this: it has attempted a construction
of those laws which, according to its foundations, is alto
gether empirical. We may assume it as a general rule,
that in any proposed construction, that which is not a
pure general form cannot have any scientific import
or truth. The foundation from which the centrifugal
motion of the bodies of the world is derived, is no ne
cessary form, it is an empirical fact. The Newtonian
attractive force, even if it be a necessary assumption for
a merely reflective view of the subject, is still of no
significance for the Reason, which recognizes only abso
lute relations. The grounds of the Keplerian laws can
be derived, without any empirical appendage, purely
from the doctrine of Ideas, and of the two Unities, which
are in themseves one Unity, and in virtue of which each
being, while it is absolute in itself, is at the same time
in the absolute, and reciprocally."
It will be observed, that in this passage our mecha
nical laws are objected to because they are not necessary
results of our ideas ; which, however, as we have seen,
according to the opinion of some eminent mechanical
philosophers, they are. But to assume this evident
necessity as a condition of every advance in science, is
to mistake the last, perhaps unattainable step, for the
first, which lies before our feet. And, without inquiring
further about " the Doctrine of the two Unities," or the
manner in which from that doctrine we may deduce the
Keplerian laws, we may be well convinced that such a
doctrine cannot supply any sufficient reason to induce us
to quit the inductive path by which all scientific truth
up to the present time has been acquired.
G. But without going to schools of philosophy oppo
sed to the Inductive School, we may find many loose and
vague habits of thinking on mechanical subjects among
the common classes of readers and reasoners. And
DIFFUSION OF CLEAR MECHANICAL IDEAS. 20*9
there are some familiar modes of employing the phrase
ology of mechanical science, which are, in a certain
degree, chargeable with inaccuracy, and may produce
or perpetuate confusion. Among such cases we may
mention the way in which the centripetal and centri
fugal forces, and also the projectile and central forces
of the planets, are often compared or opposed. Such
antitheses sometimes proceed upon the false notion that
the two members of these pairs of forces are of the
same kind : whereas on the contrary the projectile force
is a hypothetical impulsive force which may, at some
former period, have caused the motion to begin ; while
the central force is an actual force, which must act con
tinuously and during the whole time of the motion, in
order that the motion may go on in the curve. In the
same manner the centrifugal force is not a distinct force
in a strict sense, but only a certain result of the first
law of motion, measured by the portion of centripetal
force wrhich counteracts it. Comparisons of quantities
so heterogeneous imply confusion of thought, and often
suggest baseless speculations and imagined reforms of
the received opinions.
7. I might point out other 'terms and maxims, in
addition to those already mentioned, which, though for
merly employed in a loose and vague manner, are now
accurately understood and employed by all just thinkers;
and thus secure and diffuse a right understanding of me
chanical truths. Such are momentum, inertia, quantity
of matter, quantity of motion ; thai force is proportional
to its effects; that action and reaction are equal; that
what is gained in force by machinery is lost in time;
that the quantity of motion in the world cannot be either
increased or diminished. When the expression of the
truth thus becomes easy and simple, clear and con
vincing, the meanings given to words and phrases by
270 PHILOSOPHY OF THE MECHANICAL SCIENCES.
discoverers glide into the habitual texture of men's rea
sonings, and the effect of the establishment of true
mechanical principles is felt far from the school of the
mechanician. If these terms and maxims are understood
with tolerable clearness, they carry the influence of
truth to those who have no direct access to its sources.
Many an extravagant project in practical machinery, and
many a wild hypothesis in speculative physics, has been
repressed by the general currency of such maxims as we
have just quoted.
8. Indeed so familiar and evident are the elementary
truths of mechanics when expressed in this simple form,
that they are received as truisms ; and men are disposed
to look back with surprize and scorn at the speculations
which were carried on in neglect of them. The most
superficial reasoner of modern times thinks himself enti
tled to speak with contempt and ridicule of Kepler's
hypothesis concerning the physical causes of the celestial
motions: and gives himself credit for intellectual supe
riority, because he sees, as self-evident, what such a man
could not discover at all. It is well for such a person to
recollect, that the real cause of his superior insight is
not the pre-eminence of his faculties, but the successful
labours of those who have preceded him. The language
which he has learnt to use unconsciously, has been
adapted to, and moulded on, ascertained truths. When
he talks familiarly of " accelerating forces" and " de
flexions from the tangent," he is assuming that which
Kepler did not know, and which it cost Galileo and his
disciples so much labour and thought to establish. Lan
guage is often called an instrument of thought ; but it
is also the nutriment of thought ; or rather, it is the
atmosphere in which thought lives : a medium essential
to the activity of our speculative power, although invi
sible and imperceptible in its operation ; and an element
DIFFUSION OF CLEAR MECHANICAL IDEAS. '271
modifying, by its qualities and changes, the growth and
complexion of the faculties which it feeds. In this way
the influence of preceding discoveries upon subsequent
ones, of the past upon the present, is most penetrating
and universal, though most subtle and difficult to trace.
The most familiar words and phrases are connected by
imperceptible ties with the reasonings and discoveries of
former men and distant times. Their knowledge is an
inseparable part of ours ; the present generation inherits
and uses the scientific wealth of all the past. And this
is the fortune, not only of the great and rich in the
intellectual world : of those who have the key to the
ancient storehouses, and who have accumulated treasures
of their own; — but the humblest inquirer, while he
puts his reasonings into words, benefits by the labours
of the greatest discoverers. When he counts his little
wealth, he finds that he has in his hands coins which
bear the image and superscription of ancient and modern
intellectual dynasties ; and that in virtue of this posses
sion, acquisitions are in his power, solid knowledge
within his reach, which none could ever have attained
to, if it were not that the gold of truth, once dug out of
the mine, circulates more and more widely among man
kind.
9. Having so fully examined, in the preceding in
stances, the nature of the progress of thought which
science implies, both among the peculiar cultivators of
science, and in that wider world of general culture which
receives only an indirect influence from scientific disco
veries, we shall not find it necessary to go into the same
extent of detail with regard to the other provinces of
human knowledge. In the case of the Mechanical
Sciences, we have endeavoured to show, not only that
Ideas are requisite in order to form into a science the
Facts which nature otters to us, but that we can advance,
'272 PHILOSOPHY OF THE MECHANICAL SCIENCES.
almost or quite, to a complete identification of the Facts
with the Ideas. In the sciences to which we now pro
ceed, we shall not seek to fill up the chasm by which
Facts and Ideas are separated ; but we shall endeavour
to detect the Ideas which our knowledge involves, to
show how essential these arc ; and in some respects to
trace the mode in which they have been gradually de
veloped among men.
10. The motions of the heavenly bodies, their laws,
their causes, are among the subjects of the first division
of the Mechanical Sciences ; and of these sciences we
formerly sketched the history, and have now endeavoured
to exhibit the philosophy. If we were to take any other
class of motions, their laws and causes might give rise
to sciences which would be mechanical sciences in exactlv
«/
the same sense in which Physical Astronomy is so. The
phenomena of magnets, of electrical bodies, of galva-
nical apparatus, seem to form obvious materials for such
sciences ; and if they were so treated, the philosophy of
such branches of knowledge would naturally come under
our consideration at this point of our progress.
But on looking more attentively at the sciences of
Electricity, Magnetism, and Galvanism, we discover
cogent reasons for transferring them to another part of
our arrangement ; we find it advisable to associate them
with Chemistry, and to discuss their principles when
we can connect them with the principles of chemical
science. For though the first steps and narrower gene
ralizations of these sciences depend upon mechanical
ideas, the highest laws and widest generalizations which
we can reach respecting them, involve chemical rela
tions. The progress of these portions of knowledge is
in some respects opposite to the progress of Physical
Astronomy. In this, we begin with phenomena which
appear to indicate peculiar and various qualities in the
DIFFUSION OF CLEAR MECHANICAL IDEAS. 273
bodies which we consider, (namely, the heavenly bodies,)
and we find in the end that all these qualities resolve
themselves into one common mechanical property, which
exists alike in all bodies and parts of bodies. On the
contrary, in studying magnetical and electrical laws, we
appear at first to have a single extensive phenomenon,
attraction and repulsion : but in our attempts to gene
ralize this phenomenon, we find that it is governed by
conditions depending upon something quite separate
from the bodies themselves, upon the presence and dis
tribution of peculiar and transitory agencies ; and, so far
as we can discover, the general lawrs of these agencies
are of a chemical nature, and are brought into action by
peculiar properties of special substances. In cosmical
phenomena, everything, in proportion as it is referred to
mechanical principles, tends to simplicity, — to permanent
uniform forces, — to one common, positive, property. In
magnetical and electrical appearances, on the contrary,
the application of mechanical principles leads only to
a new complexity, which requires a new explanation;
and this explanation involves changeable and various
forces, — gradations and oppositions of qualities. The
doctrine of the universal gravitation of matter is a simple
and ultimate truth, in which the mind can acquiesce
and repose. We rank gravity among the mechanical
attributes of matter, and we see no necessity to derive
it from any ulterior properties. Gravity belongs to mat
ter, independent of any conditions. But the conditions of
magnetic or electrical activity require investigation as
much as the lares of their action. Of these conditions
no mere mechanical explanation can be given ; we are
compelled to take along with us chemical properties
and relations also : and thus magnetism, electricity, gal
vanism, are mechanico-chemical sciences.
11. Before considering these, therefore, I shall treat
VOL. I. W. P. T
274 PHILOSOPHY OF THE MECHANICAL SCIENCES.
of what I shall call Secondary Mechanical Sciences ; by
which expression I mean the sciences depending upon
certain qualities which our senses discover to us in
bodies; — Optics, which has visible phenomena for its
subject; Acoustics, the science of hearing; the doctrine
of Ih'ftt, a quality which our touch recognizes : to this
last science 1 shall take the liberty of sometimes giving
the name T/iermotics, analogous to the names of the
other two. If our knowledge of the phenomena of Smell
and Taste had been successfully cultivated and syste
matized, the present part of our work would be the
place for the philosophical discussion of those sensations
as the subjects of science.
The branches of knowledge thus grouped in one class
involve common Fundamental Ideas, from which their
principles are derived in a mode analogous, at least in
a certain degree, to the mode in which the principles of
the mechanical sciences are derived from the funda
mental ideas of causation and reaction. We proceed
now to consider these Fundamental Ideas, their nature,
development, and consequences.
ADDITIONAL NOTE TO CHAPTER IV.— ON THE AXIOMS
WHICH RELATE TO THE IDEA OF CAUSE.
THE Axiom that Reaction is equal and opposite to Action^ may appear
to be at variance with a maxim concerning Cause which is commonly
current ; namely, that the " Cause precedes Effect, and Effect follows
Cause." For it may he said, if J., the Action, and -R, the Reaction, can
be considered as mutually the cause of each other, A must precede R,
and yet must follow it, which is impossible. But to this I reply, that
in those cases of direct Causation to which the maxim applies, the Cause
and Effect are not successive, but simultaneous. If I press against some
obstacle, the obstacle resists and returns the pressure at the instant it is
exerted, not after any interval of time, however small. The common
NOTES ON CHAPTERS IV. AND VI. 275
maxim, that the effect follows the cause, has arisen from the practice of
considering, as examples of cause and effect, not instantaneous forces or
causes, and the instantaneous changes which they produce ; hut taking,
instead of this latter, the cumulative effects produced in the course of
time, and compared with like results occurring without the action of the
cause. Thus, if we alter the length of a clock-pendulum, this change
produces, as its effect, a subsequent change of rate in the clock : because
the rate is measured by the accumulated effects of the pendulum's gravity,
before and after the change. But the pendulum produces its mechanical
effect upon the escapement, at the moment of its contact, and each
wheel upon the next, at the moment of its contact. As has been said
in a Review of this work, " The time lost in cases of indirect physical
causation is consumed in the movements which take place among the
parts of the mechanism in action, by which the active forces so trans
formed into momentum are transported over intervals of space to new
points of action, the motion of matter in such cases being regarded as a
mere carrier of force." (Quarterly Rev., No. cxxxv., p. 212.) See this
subject further treated in a Memoir entitled, " Discussion of the Ques
tion : — Are Cause and Effect Successive or Simultaneous ?" in the
Memoirs of the Cambridge Philosophical Society, Vol. vii. Part Hi.
ADDITIONAL NOTE TO CHAPTER VI., SECT. 5— ON
THE CENTER OF GRAVITY.
To the doctrine that mechanical principles, such as the one here under
consideration (that the pressure on the point of support is equal to the
sum of the weights), are derived from our Ideas, and do not flow from
but regulate our experience, objections are naturally made by those who
assert all our knowledge to be derived from experience. How, they ask,
can. we know the properties of pressures, levers and the like, except
from experience? What but experience can possibly inform us that a
force applied transversely to a lever will have any tendency to tuni the
lever on its center ? This cannot be, except we suppose in the lever
tenacity, rigidity and the like, which are qualities known only by
experience. And it is obvious that this line of argument might be
carried on through the whole subject.
My answer to this objection is a remark of the same kind as one
which I have made respecting the Ideas of Space, Time, and Number,
in a Note at the end of Chapter x. of the last Book. The mind, in
apprehending events as causes and effects, is govenied by Laws of its
own Activity ; and these Laws govern the results of the mind's action ;
T2
276 PHILOSOPHY OF THE MECHANICAL SCIENCES.
and make these results conform to the Axioms of Causation. But this
activity of the mind is awakened and developed by being exercised ;
and in dealing with the examples of cause and effect here spoken of,
(namely, pressure and resistance, force and motion,) the mind's activity
is necessarily governed also by the bodily powers of perception and
action. We are human beings only in so far as we have existed in space
and time, and of our human faculties, developed by our existence in space
and time, space and time are necessary conditions. In like manner, we
are human beings only in so far as we have bodies, and bodily organs ;
and our bodies necessarily imply material objects external to us. And
hence our human faculties, developed by our bodily existence in a
material world, have the conditions of matter for their necessary Laws.
I have already said (Chap, v.) that our conception of Force arises with
our consciousness of our own muscular exertions ; — that Force cannot
be conceived without Resistance to exercise itself upon ; — and that this
resistance is supplied by Matter. And thus the conception of Matter,
and of the most general modes in which Matter receives, resists, and
transmits force, are parts of our constitution which, though awakened
and unfolded by our being in a material world, are not distinguishable
from the original structure of the mind. I do not ascribe to the
mind Ideas which it would have, even if it had no intercourse with
the world of space, time, and matter; because we cannot imagine a
mind in such a state. But I attempt to point out and classify those
Conditions of all Experience, to which the intercourse of all minds with
the material world has necessarily given rise in all. Truths thus neces
sarily acquired in the course of all experience, cannot be said to be
learnt from experience, in the same sense in which particular facts, at
definite times, are learnt from experience, learnt by some persons and
not by others, learnt with more or less of certainty. These latter
special truths of experience will be very important subjects of our con
sideration; but our whole chance of discussing them with any profit
depends upon our keeping them distinct from the necessary and uni
versal conditions of experience. Here, as everywhere, we must keep
in view the fundamental antithesis of Ideas and Facts.
277
BOOK IV.
THE PHILOSOPHY OF THE SECONDARY
MECHANICAL SCIENCES.
CHAPTER I.
OF THE IDEA OF A MEDIUM AS COMMONLY
EMPLOYED.
1 . Of Primary and Secondary Qualities. — IN the
same way in which the mechanical sciences depend upon
the Idea of Cause, and have their principles regulated
by the development of that Idea, it will be found that
the sciences which have for their subject Sound, Light,
and Heat, depend for their principles upon the Funda
mental Idea of Media by means of which we perceive
those qualities. Like the idea of cause, this idea of a
medium is unavoidably employed, more or less distinctly,
in the common, unscientific operations of the under
standing; and is recognized as an express principle in
the earliest speculative essays of man. But here also,
as in the case of the mechanical sciences, the develope-
ment of the idea, and the establishment of the scientific
truths which depend upon it, was the business of a
succeeding period, and was only executed by means of
long and laborious researches, conducted with a constant
reference to experiment and observation.
Among the most prominent manifestations of the
influence of the idea of a medium of which we have
now to speak, is the distinction of the qualities into
278 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
primary, and secondary qualities. This distinction has
been constantly spoken of in modern times : yet it has
often been a subject of discussion among metaphysicians
whether there be really such a distinction, and what the
true difference is. Locke states it thus* : original or
primary qualities of bodies are " such as are utterly in
separable from the body in what estate soever it may
be, — such as sense constantly finds in every particle of
matter which has bulk enough to be perceived, and the
mind finds inseparable from every particle of matter,
though less than to make itself singly perceived by our
senses :" and he enumerates them as solidity, extension,
figure, motion or rest, and number. Secondary qualities,
on the other hand, are such "which in truth are nothing
in the objects themselves, but powers to produce various
sensations in us by their primary qualities, i. e., by the
bulk, figure, texture, and motion of their insensible
parts, as colours, sounds, tastes, &c."
Dr. Reidf, reconsidering this subject, puts the differ
ence in another way. There is, he says, a real foundation
for the distinction of primary and secondary qualities,
and it is this : " That our senses give us a direct and dis
tinct notion of the primary qualities, and inform us what
they are in themselves ; but of the secondary qualities,
our senses give us only a relative and obscure notion.
They inform us only that they are qualities that affect us
in a certain manner, that is, produce in us a certain sen
sation ; but as to what they are in themselves, our senses
leave us in the dark."
Dr. Brown | states the distinction somewhat other
wise. We give the name of matter, he observes, to that
which has extension and resistance : these, therefore, are
primary qualities of matter, because they compose our
* Essa//, B. n. ch. viii, s. 9, 10, t Essays, B. n. c. xvii.
.| Lectures, n. 12.
OF THE IDEA OF A MEDIUM. 279
definition of it. All other qualities are secondary, since
they are ascribed to bodies only because we find them
associated with the primary qualities which form our
notion of those bodies.
It is not necessary to criticize very strictly these vari
ous distinctions. If it were, it would be easy to find
objections to them. Thus Locke, it may be observed,
does not point out any reason for believing that his
secondary qualities are produced by the primary. How
are we to learn that the colour of a rose arises from the
bulk, figure, texture, and motion of its particles ? Cer
tainly our senses do not teach us this; and in what other
way, on Locke's principles, can we learn it? Reid's
statement is not more free from the same objection.
How does it appear that our notion of Warmth is rela
tive to our own sensations more than our notion of
Solidity ? And if we take Brown's account, we may still
ask whether our selection of certain qualities to form
our idea and definition of matter be arbitrary and with
out reason? If it be, how can it make a real distinction?
if it be not, what is the reason ?
I do not press these objections, because I believe that
any of the above accounts of the distinction of primary
and secondary qualities is right in the main, however
imperfect it may be. The difference between such
qualities as Extension and Solidity on the one hand,
and Colour or Fragrance on the other, is assented to
by all, with a conviction so firm and indestructible, that
there must be some fundamental principle at the bottom
of the belief, however difficult it may be to clothe the
principle in words. That successive efforts to express
the real nature of the difference were made by men so
clear-sighted and acute as those whom I have quoted,
even if none of them are satisfactory, shows how strong
and how deeply-seated is the perception of truth which
impels us to such attempts.
280 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
The most obvious mode of stating the difference of
primary and secondary qualities, as it naturally offers
itself to speculative minds, appears to be that employed
bv Locke, slightly modified. Certain of the qualities of
bodies, as their bulk, figure, and motion, are perceived
immediately in the bodies themselves. Certain other
qualities as sound, colour, heat, are perceived by means
of some medium. Our conviction that this is the case
is spontaneous and irresistible ; and this difference of
qualities immediately and mediately perceived is the dis
tinction of primary and secondary qualities. We proceed
further to examine this conviction.
2. The Idea of Externality. — In reasoning concern
ing the secondary qualities of bodies, we are led to assume
the bodies to be external to us, and to be perceived by
means of some medium intermediate between us and
them. These assumptions are fundamental conditions
of perception, inseparable from it even in thought.
That objects are external to us, that they are without
us, that they have outness, is as clear as it is that these
words have any meaning at all. This conviction is, in
deed, involved in the exercise of that faculty by which
we perceive all things as existing in space ; for by this
faculty we place ourselves and other objects in one com
mon space, and thus they are exterior to us. It may be
remarked that this apprehension of objects as external
to us, although it assumes the idea of space, is far from
being implied in the idea of space. The objects which
we contemplate are considered as existing in space, and
by that means become invested with certain mutual rela
tions of position ; but when we consider them as existing
without us, we make the additional step of supposing
ourselves and the objects to exist in one common space.
The question respecting the Ideal Theory of Berkeley has
been mixed up with the recognition of this condition of
the externality of objects. That philosopher maintained,
OF THE IDEA OF A MEDIUM. 281
as is well known, that the perceptible qualities of bodies
have no existence except in a perceiving mind. This
system has often been understood as if he had imagined
the world to be a kind of optical illusion, like the images
which we see when we shut our eyes, appearing to be
without us, though they are only in our organs ; and
thus this Ideal System has been opposed to a belief in
an external world. In truth, however, no such opposi
tion exists. The Ideal System is an attempt to explain
the mental process of perception, and to get over the
difficulty of mind being affected by matter. But the
author of that system did not deny that objects were
perceived under the conditions of space and mechanical
causation ; — that they were external and material so far
as those words describe perceptible qualities. Berkeley's
system, however visionary or erroneous, did not prevent
his entertaining views as just, concerning optics or acous
tics, as if he had held any other doctrine of the nature
of perception.
But when Berkeley's theory was understood as a
denial of the existence of objects without us, how was it
answered ? If we examine the answers which are given
by Reid and other philosophers to this hypothesis, it will
be found that they amount to this : that objects are
without us, since we perceive that they are so ; that we
perceive them to be external, by the same act by which
we perceive them to be objects. And thus, in this stage
of philosophical inquiry, the externality of objects is re
cognized as one of the inevitable conditions of our per
ception of them ; and hence the Idea of Externality is
adopted as one of the necessary foundations of all rea
soning concerning all objects whatever.
3. Sensation by a Medium. — Objects, as we have just
seen, are necessarily apprehended as without us ; and in
general, as removed from us by a great or small distance.
282 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
Yet they affect our bodily senses ; and this leads us ir
resistibly to the conviction that they are perceived by
means of something intermediate. Vision, or hearing,
or smell, or the warmth of a fire, must be communicated
to us by some medium of sensation. This unavoidable
belief appears in all attempts, the earliest and the latest
alike, to speculate upon such subjects. Thus, for in
stance, Aristotle says *, " Seeing takes place in virtue of
some action which the sentient organ suffers : now it
cannot suffer action from the colour of the object di
rectly : the only remaining possible case then is, that it
is acted upon by an intervening Medium ; there must
then be an intervening Medium." " And the same may
be said," he adds, " concerning sounding and odorous
bodies ; for these do not produce sensation by touching
the sentient organ, but the intervening Medium is acted
on by the sound or the smell, and the proper organ, by
the Medium.. ..In sound the Medium is air; in smell we
have no name for it." In the sense of taste, the neces
sity of a Medium is not at first so obviously seen, because
the object tasted is brought into contact with the organ ;
but a little attention convinces us that the taste of a
solid body can only be perceived when it is conveyed
in some liquid vehicle. Till the fruit is crushed, and
till its juices are pressed out, we do not distinguish its
flavour. In the case of heat, it is still more clear that
we are compelled to suppose some invisible fluid, or
other means of communication, between the distant body
which warms us and ourselves.
It may appear to some persons that the assumption
of an intermedium between the object perceived and the
sentient organ results from the principles which form
the basis of our mechanical reasonings, — that every
change must have a cause, and that bodies can act upon
* nc/)'( Vvx^- IL 7-
OF THE IDEA OF A MEDIUM. 283
each other only by contact. It cannot be denied that
this principle does offer itself very naturally as the
ground of our belief in media of sensation ; and it appears
to be referred to for this purpose by Aristotle in the
passage quoted above. But yet we cannot but ask,
Does the principle, that matter produces its effect by
contact only, manifestly apply here ? When we so apply
it, we include sensation among the effects which material
contact produces ; — a case so different from any merely
mechanical effect, that the principle, so employed, ap
pears to acquire a new signification. May we not, then,
rather say that we have here a new axiom, — That sensa
tion implies a material cause immediately acting on the
organ, — than a new application of our former proposi
tion, — That all mechanical change implies contact ?
The solution of this doubt is not of any material con
sequence to our reasonings ; for whatever be the ground
of the assumption, it is certain that we do assume the
existence of media by which the sensations of sight,
hearing, and the like, are produced ; and it will be seen
shortly that principles inseparably connected with this
assumption are the basis of the sciences now before us.
This assumption makes its appearance in the physical
doctrines of all the schools of philosophy. It is ex
hibited perhaps most prominently in the tenets of the
Epicureans, who were materialists, and extended to all
kinds of causation the axiom of the existence of a cor
poreal mechanism by which alone the effect is produced.
Thus, according to them, vision is produced by certain
images or material films which flow from the object,
strike upon the eyes, and so become sensible. This
opinion is urged with great detail and earnestness by
Lucretius, the poetical expositor of the Epicurean creed
among the Romans. His fundamental conviction of the
necessity of a material medium is obviously the basis of
284 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
his reasoning, though he attempts to show the existence
of such a medium by facts. Thus he argues'-, that by
shouting loud we make the throat sore ; which shows,
he says, that the voice must be material, so that it can
hurt the passage in coming out.
Hand igitur dubium est quin voces verbaque constcnt
Corporeis e principiis lit laxlere possint
4. The Process of Perception of Secondary Quali
ties, — The likenesses or representatives of objects by
which they affect our senses were called by some writers
species, or sensible species, a term which continued in
use till the revival of science. It may be observed that
the conception of these species as films cast off from the
object, and retaining its shape, was different, as we have
seen, from the view which Aristotle took, though it has
sometimes been called the Peripatetic doctrine f. We
may add that the expression was latterly applied to
express the supposition of an emanation of any kind, and
implied little more than that supposition of a medium
of which we are now speaking. Thus Bacon, after re
viewing the phenomena of sound, says \, " Videntur
motus soni fieri per species spirituales : ita enim loquen-
dum donee certius quippiam inveniatur."
Though the fundamental principles of several sciences
depend upon the assumption of a medium of perception,
these principles do not at all depend upon any special
view of the process of our perceptions. The mechanism
of that process is a curious subject of consideration ; but
it belongs to physiology, more properly than either to
metaphysics, or to those branches of physics of which we
are now speaking. The general nature of the process is
the same for all the senses. The object affects the ap
propriate intermedium ; the medium, through the proper
* Lib. iv. 529 t Brown, Vol. 11. p. 98.
J Hist. Son. ct And., Vol. ix. p. 87.
OF THE IDEA OF A MEDIUM.
organ, the eye, the ear, the nose, affects the nerves of
the particular sense ; and, by these, in some way, the
sensation is conveyed to the mind. But to treat the
impression upon the nerves as the act of sensation which
we have to consider, would be to mistake our object,
which is not the constitution of the human body, but of
the human mind. It would be to mistake one link for
the power which holds the end of the chain. No anato
mical analysis of the corporeal conditions of vision, or
hearing, or feeling warm, is necessary to the sciences of
Optics, or Acoustics, or Thermotics.
Not only is this physiological research an extraneous
part of our subject, but a partial pursuit of such a
research may mislead the inquirer. We perceive objects
by means of certain media, and by means of certain
impressions on the nerves : but we cannot with pro
priety say that we perceive either the media or the
impressions on the nerves. What person in the act of
seeing is conscious of the little coloured spaces on the
retina? or of the motions of the bones of the auditory
apparatus whilst he is hearing? Surely, no one. This
may appear obvious enough, and yet a writer of no
common acuteness, Dr. Brown, has put forth several
very strange opinions, all resting upon the doctrine that
the coloured spaces on the retina are the objects which
we perceive; and there are some supposed difficulties
and paradoxes on the same subject which have become
quite celebrated (as upright vision with inverted images),
arising from the same confusion of thought.
As the consideration of the difficulties which have
arisen respecting the philosophy of perception may serve
still further to illustrate the principles on which we
necessarily reason respecting the secondary qualities of
bodies, I shall here devote a few pages to that subject.
280
CHAPTER II.
ON PECULIARITIES IN THE PERCEPTIONS OF
THE DIFFERENT SENSES.
1. WE cannot doubt that we perceive all secondary
qualities by means of immediate impressions made,
through the proper medium of sensation, upon our
organs. Hence all the senses are sometimes vaguely
spoken of as modifications of the sense of feeling. It
will, however, be seen, on reflection, that this mode of
speaking identifies in words things which in our concep
tions have nothing in common. No impression on the
organs of touch can be conceived as having any resem
blance to colour or smell. No effort, no ingenuity, can
enable us to describe the impressions of one sense in
terms borrowed from another.
The senses have, however, each its peculiar powers,
and these powers may be in some respects compared, so
as to show their leading resemblances and differences,
and the characteristic privileges and laws of each. This
is what we shall do as briefly as possible.
SECT. I. — Prerogatives of Sight.
THE sight distinguishes colours, as the hearing distin
guishes tones ; the sight estimates degrees of brightness,
the ear, degrees of loudness ; but with several resem
blances, there are most remarkable differences between
these two senses.
2. Position. — The sight has this peculiar prerogative,
that it apprehends the place of its objects directly and
primarily. We see where an object is at the same in
stant that we see what it is. If we see two objects, we
see their relative position. We cannot help perceiving
PECULIARITIES OF THE PERCEPTIONS. 287
that one is above or below, to the right or to the left of
the other, if we perceive them at all.
There is nothing corresponding to this in sound.
When we hear a noise, we do not necessarily assign a
place to it. It may easily happen that we cannot tell
from which side a thunder-clap comes. And though we
often can judge in what direction a voice is heard, this is
a matter of secondary impression, and of inference from
concomitant circumstances, not a primary fact of sensa
tion. The judgments which we form concerning the
position of sounding bodies are obtained by the con
scious or unconscious comparison of the impressions
made on the two ears, and on the bones of the head in
general ; they are not inseparable conditions of hearing.
We may hear sounds, and be uncertain whether they are
" above, around, or underneath !" but the moment any
thing visible appears, however unexpected, we can say,
" see where it comes !"
Since we can see the relative position of things, we
can see figure, which is but the relative position of the
different parts of the boundary of the object. And thus
the whole visible world exhibits to us a scene of various
shapes, coloured and shaded according to their form and
position, but each having relations of position to all the
rest ; and altogether, entirely filling up the whole range
which the eye can command.
3. Distance. — The distance of objects from us is no
matter of immediate perception, but is a judgment and
inference formed from our sensations, in the same way
as our judgment of position by the ear. That this is so,
was most distinctly shown by Berkeley, in his New
Theory of Vision. The elements on which we form our
judgment are, the effort by which we fix both eyes on
the same object, the effort by which we adjust each eye
to distinct vision, and the known forms, colours, and
288 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
parts of objects, as compared with their appearance.
The right interpretation of the information which these
circumstances give us respecting the true distances and
forms of things, is gradually learnt by experience, the
lesson being begun in our earliest infancy, and incul
cated upon us every hour during which we use our eyes.
The completeness with which the lesson is learnt is
truly admirable ; for we forget that our conclusion is
obtained indirectly, and mistake a judgment on evidence
for an intuitive perception. This, however, is not more
surprizing than the rapidity and unconsciousness of effort
with which we understand the meaning of the speech
that we hear, or the book that we read. In both cases,
the habit of interpretation is become as familiar as the
act of perception. And this is the case with regard to
vision. We see the breadth of the street as clearly and
readily as we see the house on the other side of it. We
see the house to be square, however obliquely it be pre
sented to us. Indeed the difficulty is, to recover the
consciousness of our real and original sensations ; — to
discover what is the apparent relation of the lines which
appear before us. As we have already said, in the com
mon process of vision we suppose ourselves to see that
which cannot be seen ; and when we would make a
picture of an object, the difficulty is to represent what is
visible and no more.
But perfect as is our habit of interpreting what we
perceive, we could not interpret if we did not perceive.
If the eye did not apprehend visible position, it could
not infer actual position, which is collected from visible
position as a consequence : if we did not see apparent
figure, we could not arrive at any opinion concerning
real form. The perception of place, which is the prero
gative of the eye, is the basis of all its other superiority.
The precision with which the eye can judge of appa-
PECULIARITIES OF TIIK PERCEPTIONS.
rent position is remarkable. If we had before us two
stars distant from each other by one-twentieth of the
moon's diameter, we could easily decide the apparent
direction of the one from the other, as above or below,
to the right or left. Yet eight millions of stars might be
placed in the visible hemisphere of the sky at such dis
tances from each other ; and thus the eye would recog
nize the relative position in a portion of its range not
greater than one eight-millionth of the whole. Such is
the accuracy of the sense of vision in this respect ; and,
indeed, we might with truth have stated it much higher.
Our judgment of the position of distant objects in a
landscape depends upon features far more minute than
the magnitude we have here described.
As our object is to point out principally the differ
ences of the senses, we do not dwell upon the delicacy
with which we distinguish tints and shades, but proceed
to another sense.
SECT. II. — Prerogatives of Hearing.
THE sense of hearing has two remarkable prerogatives ;
it can perceive a definite and peculiar relation between
certain tones, and it can clearly perceive two tones to
gether; in both these circumstances it is distinguished
from vision, and from the other senses.
4. Musical Internals. — We perceive that two tones
have, or have not, certain definite relations to each
other, which we call Concords : one sound is a Fifth, an
Octave, &c., above the other. And when this is the case,
our perception of the relation is extremely precise. It
is easy to perceive when a fifth is out of tune by one-
twentieth of a tone ; that is, by one-seventieth of itself.
To this there is nothing analogous in vision. Colours
have certain vague relations to one another; they look
well together, by contrast or by resemblance ; but
VOL. i. \v. P. LT
290 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
is an indefinite, and in most cases a casual and variable
feeling. The relation of complementary colours to one
another, as of red to green, is somewhat more definite ;
but still, has nothing of the exactness and peculiarity
which belongs to a musical concord. In the case of the
two sounds, there is an exact point at which the relation
obtains ; when by altering one note we pass this point,
the concord does not gradually fade away, but instantly
becomes a discord ; and if we go further still, we obtain
another concord of quite a different character.
We learn from the theory of sound that concords
occur when the times of vibration of the notes have
exact simple ratios; an octave has these times as 1 to 2;
a fifth, as 2 to 3. According to the undulatory theory
of light, such ratios occur in colours, yet the eye is not
affected by them in any peculiar way. The times of the
undulations of certain red and certain violet rays are
as 2 to 3, but we do not perceive any peculiar harmony
or connexion between those colours.
5. Chords. — Again, the ear has this prerogative, that
it can apprehend two notes together, yet distinct. If
two notes, distant by a fifth from each other, are sounded
on two wind instruments, both they and their musical
relation are clearly perceived. There is not a mixture,
but a concord, an interval. In colours, the case is other
wise. If blue and yellow fall on the same spot, they
form green ; the colour is simple to the eye ; it can no
more be decomposed by the vision than if it were the
simple green of the prismatic spectrum : it is impossible
for us, by sight, to tell whether it is so or not.
These are very remarkable differences of the two
senses : two colours can be compounded into an appa
rently simple one ; two sounds cannot : colours pass into
each other by gradations and intermediate tints ; sounds
pass from one concord to another by no gradations : the
PECULIARITIES OF THE PERCEPTIONS. 201
most intolerable discord is that which is near a concord.
We shall hereafter see how these differences affect the
scales of sound and of colour.
6. Rhythm. — We might remark, that as we see ob
jects in space, we hear sounds in time ; and that we thus
introduce an arrangement among sounds which has
several analogies with the arrangement of objects in
space. But the conception of time does not seem to be
peculiarly connected with the sense of hearing; a faculty
of apprehending tone and time, or in musical phrase
ology tune and rhythm, are certainly very distinct. I
shall not, therefore, here dwell upon such analogies.
The other Senses have not any peculiar prerogatives,
at least none which bear on the formation of science. I
may, however, notice, in the feeling of heat, this cir
cumstance ; that it presents us with two opposites, heat
and cold, which graduate into each other. This is not
quite peculiar, for vision also exhibits to us white and
black, which are clearly opposites, and which pass into
each other by the shades of gray.
SECT. III. — The Paradoxes of Vision.
1. First Paradox of Vision. Upright Vision.—
All our senses appear to have this in common ; — That
they act by means of organs, in which a bundle of nerves
receives the impression of the appropriate medium of
the sense. In the construction of these organs there are
great differences and peculiarities, corresponding, in part
at least, to the differences in the information given.
Moreover, in some cases, as we have noted in the case of
audible position and visible distance, that which seems
to be a perception is really a judgment founded on per
ceptions of which we are not directly aware. It will be
seen, therefore, that with respect to the peculiar powers
of each sense, it may be asked ; — whether they can be
u 2
29*2 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
explained by the construction of the peculiar organ ;—
whether they are acquired judgments arid not direct
perceptions ; — or whether they are inexplicable in either
of these ways, and cannot, at present at least, be re
solved into anything but conditions of the intellectual
act of perception.
Two of these questions with regard to vision, have
been much discussed by psychological writers: the cause
of our seeing objects upright by inverted images on
the retina; and of our seeing single with two such
images.
Physiologists have very completely explained the
exquisitely beautiful mechanism of the eye, considered
as analogous to an optical instrument ; and it is in
disputable that by means of certain transparent lenses
and humours, an inverted image of the objects which
are looked at is formed upon the retina, or fine net
work of nerve, with which the back of the eye is lined.
We cannot doubt that the impression thus produced on
these nerves is essential to the act of vision ; and so far
as we consider the nerves themselves to feel or perceive
by contact, we may say that they perceive this image,
or the affections of light which it indicates. But we
cannot with any propriety say that me perceive, or that
our mind perceives, this image; for we are not conscious
of it, and none but anatomists are aware of its existence:
we perceive by means of it.
A difficulty has been raised, and dwelt upon in a
most unaccountable manner, arising from the neglect of
this obvious distinction. It has been asked, how is it
that we see an object, a man for instance, upright, when
the immediate object of our sensation, the image of the
man on our retina, is inverted ? To this we must answer,
that we see him upright because the image is inverted ;
that the inverted image is the necessary means of seeing
PECULIARITIES OF THE PERCEPTIONS. 293
an upright object. This is granted, and where then is
the difficulty? Perhaps it may be put thus: How is it
that we do not judge the man to be inverted, since the
sensible image is so ? To this we may reply, that we
have no notion of upright or inverted, except that. which
is founded on experience, and that all our experience,
without exception, must have taught us that such a
sensible image belongs to a man who is in an upright
position. Indeed, the contrary judgment is not con
ceivable ; a man is upright whose head is upwards and
his feet downwards. But what are the sensible images
of upwards asiddonmivardsf' Whatever be our standard
of up and down, the sensible representation of up will be
an image moving on the retina towards the lower side,
and the sensible representation of down will be a motion
towards the upper side. The head of the man's image is
towards the image of the sky, its feet are towards the
image of the ground ; how then should it appear other
wise than upright ? Do we expect that the whole world
should appear inverted ? Be it so : but if the whole be
inverted, how is the relation of the parts altered ? Do
we expect that we should think our own persons in par
ticular inverted ? This cannot be, for we look at them
as we do at other objects. Do we expect that things
should appear to fall upwards ? Surely not. For what
do we know of upwards, except that it is the direction
in which bodies do not fall? In short, the whole of
this difficulty, though it has in no small degree embar
rassed metaphysicians, appears to result from a very
palpable confusion of ideas ; from an attempt at com
parison of what we see, with that which the retina feels,
as if they were separately presentable. It is a sufficient
explanation to say, that we do not see the image on the
retina, but see by means of it. The perplexity does not
require much more skill to disentangle, than it does
294 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
to see -that a word written in black ink, may signify
white*.
8. Second Paradox of Vision. Single Vision. —
(1.) Small or Distant Objects. — The other difficulty, why
with two images on the retina we see only one object,
is of a much more real and important kind. This effect
is manifestly limited by certain circumstances of a very
precise nature ; for if we direct our eyes at an object
which is very near the eye, we see all other objects
double. The fact is not, therefore, that we are incapable
of receiving two impressions from the two images, but
that, under certain conditions, the two impressions form
one. A little attention shows us that these conditions
are, that with both eyes we should look at the same
object ; and again, we find that to look at an object with
either eye, is to direct the eye so that the image falls
* The explanation of our seeing objects erect when the image is
inverted lias been put very simply, by saying, " We call that the lower
end of an object which is next the ground." The observer cannot look
into his own eye ; he knows by experience what kind of image cor
responds to a man in an upright position. The anatomist tells him that
this image is inverted : but this does not disturb the process of judging
by experience. It does not appear why any one should be perplexed at
the notion of seeing objects erect by means of inverted images, rather
than at the notion of seeing objects large by means of small images ; or
cubical and pyramidal, by means of images on a spherical surface ; or
green and red, by means of images on a black surface. Indeed some
persons have contrived to perplex themselves with these latter questions,
as well as the first.
The above explanation is not at all affected, as to its substance, if we
adopt Sir David Brewstcr's expression, and say that the line of visible
direction is a line passing through the center of the spherical surface of
the retina, and therefore of course perpendicular to the surface. In
speaking of ': the inverted image," it has always been supposed to be
determined by such lines ; and though the point where they intersect
may not have been ascertained with exactness by previous physiologists,
the philosophical view of the matter was not in any degree vitiated
by this imperfection.
PECULIARITIES OF THE PERCEPTIONS. 295
on or near a particular point about the middle of the
retina. Thus these middle points in the two retinas
correspond, and we see an image single when the two
images fall on the corresponding points.
Again, as each eye judges of position, and as the two
eyes judge similarly, an object will be seen in the same
place by one eye and by the other, when the two images
which it produces are similarly situated with regard to
the corresponding points of the retina*.
This is the Law of Single Vision, at least so far as
regards small objects; namely, objects so small that in
contemplating them we consider their position only, and
not their solid dimensions. Single vision in such cases
is a result of the law of vision simply : and it is a
mistake to call in, as some have done, the influence of
* The explanation of single vision with two eyes may be put in
another form. Each eye judges immediately of the relative position of
all objects within the field of its direct vision. Therefore when we look
with both eyes at a distant prospect (so distant that the distance
between the eyes is small in comparison) the two prospects, being simi
lar collections of forms, will coincide altogether, if a corresponding point
m one and in the other coincide. If this be the case, the two images
of every object will fall upon corresponding points of the retina, and
will appear single.
If the two prospects seen by the two eyes do not exactly coincide,
in consequence of nearness of the objects, or distortion of the eyes, but
if they nearly coincide, the stronger image of an object absorbs the
weaker, and the object is seen single ; yet modified by the combination,
as will be seen when we speak of the siflgle vision of near objects.
When the two images of an object are considerably apart, we see it
double.
This explanation is not different in substance from the one given in
the text ; but perhaps it is better to avoid the assertion that the law of
corresponding points is " a distinct and original principle of our consti
tution," as I had stated in the first edition. The simpler mode of
stating the law of OUT constitution appears to be to say, that each eye
determines similarly the position of objects ; and that when the positions
of an object, as seen by the two eyes, coincide (or nearly coincide) the
object is seen single.
•2!X) PHILOSOPHY OF SKroNDANY MECHANICAL SCIENCES.
habit and of acquired judgments, in order to determine
the result in such cases.
To ascribe the apparent singleness of objects to the
impressions of vision corrected by the experience of
touch'", would be to assert that a person who had not
been in the habit of handling what he saw, would see all
objects double ; and also, to assert that a person begin
ning1 with the double world which vision thus offers to
O
him, would, by the continued habit of handling objects,
gradually and at last learn to see them single. But
all the facts of the case show such suppositions to be
utterly fantastical. No one can, in this case, go back
from the habitual judgment of the singleness of objects,
to the original and direct perception of their doubleness,
as the draughtsman goes back from judgments to per-
peption, in representing solid distances and forms by
means of perspective pictures. No one can point out
any case in which the habit is imperfectly formed ; even
children of the most tender age look at an object with
both eyes, and see it as one.
In cases when the eyes arc distorted (in squinting),
one eye only is used, or if both are employed, there is
double vision ; and thus any derangement of the corre
spondence of motion in the two eyes will produce double-
sightedness.
Brown is one of those t who assert that two images
suggest a single object because we have always found
two images to belong to a single object. He urges as
an illustration, that the two words " he conquered,"
by custom excite exactly the same notion as the one
Latin word "vicit;" and thus that two visual images,
by the effect of habit, produce the same belief of a
single object as one tactual impression. But in order
to make this pretended illustration of any value, it ought
* Sec Brown, Vol n. ]<. ol I Lectures, Vol. n. p. 81.
PECULIARITIES OF THE PERCEPTIONS. 297
to he true that when a person has thoroughly learnt
the Latin language, he can no longer distinguish any
separate meaning in " he" and in " conquered." We can
by no effort perceive the double sensation, when we
look at the object with the two eyes. Those who squint,
learn by habit to see objects single: but the habit which
they acquire is that of attending to the impressions of
one eye only at once, not of combining the two impres
sions. It is obvious, that if each eye spreads before us
the same visible scene, with the same objects and the
same relations of place, then, if one object in each scene
coincide, the whole of the two visible impressions will be
coincident. And here the remarkable circumstance is,
that not only each eye judges for itself of the relations
of position which come within its field of view; but that
there is a superior and more comprehensive faculty
which combines and compares the two fields of view;
which asserts or denies their coincidence ; which con
templates, as in a relative position to one another, these
two visible worlds, in which all other relative position is
given. This power of confronting two sets of visible
images and figured spaces before a purely intellectual
tribunal, is one of the most remarkable circumstances in
the sense of vision.
9. (2.) Near Objects. — We have hitherto spoken
of the singleness of objects whose images occupy corre
sponding positions on the retina of the two eyes. But
here occurs a difficulty. If an object of moderate size, a
small thick book for example, be held at a little dis
tance from the eyes, it produces an image on the retina
of each eye; and these two images are perspective
representations of the book from different points of view,
(the positions of the two eyes,) and are therefore of dif
ferent forms. Hence the two images cannot occupy cor
responding points of the retina throughout their whole
PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
extent. If the central parts of the two images occupy
corresponding points, the boundaries of the two will
not correspond. How is it then consistent with the
law above stated, that in this case the object appears
single ?
It may be observed, that the two images in such a case
will differ most widely when the object is not a mere sur
face, but a solid. If a book, for example, be held with
one of its upright edges towards the face, the right eye
will see one side more directly than the left eye, and
the left eye will see another side more directly, and the
outline of the two images upon the two retinas will ex
hibit this difference. And it may be further observed,
that this difference in the images received by the two
eyes, is a plain and demonstrative evidence of the solidity
of the object seen ; since nothing but a solid object
could (without some special contrivance) produce these
different forms of the images in the two eyes.
Hence the absence of exact coincidence in the two
images on the retina is the necessary condition of the
solidity of the object seen, and must be one of the indi
cations by means of which our vision apprehends an
object as solid. And that this is so, Mr. Wheatstone
has proved experimentally, by means of some most
ingenious and striking contrivances. He has devised*
an instrument by which two images (drawn in outline)
differing exactly as much as the two images of a solid
body seen near the face would differ, are conveyed,
one to one eye, and the other to the other. And it is
found that when this is effected, the object which the
images represent is not only seen single, but is appre
hended as solid with a clearness and reality of conviction
<|uitc distinct from any impression which a mere per
spective representation can give.
; Phil. Trans.. 1H3H.
PECULIARITIES OF THE PERCEPTIONS.
At the same time it is found that the object is then
only apprehended as single when the two images are
such as are capable of being excited by one single object
placed in solid space, and seen by the two eyes. If
the images differ more or otherwise than this condition
allows, the result is, that both are seen, their lines cross
ing and interfering with one another.
It may be observed, too, that if an object be of such
large size as not to be taken in by a single glance of the
eyes, it is no longer apprehended as single by a direct
act of perception ; but its parts are looked at separately
and successively, and the impressions thus obtained are
put together by a succeeding act of the mind. Hence
the objects which are directly seen as solid, will be of
moderate size ; in which case it is not difficult to show
that the outlines of the two images will differ from each
other only slightly.
Hence we are led to the following, as the Law of
Single Vision for near objects : — When the two images
in the two eyes are situated (part for part) nearly, but
not exactly, upon corresponding points, the object is ap
prehended as single, if the two images are such as are
or would be given by a single solid object seen by the
two eyes separately : and in this case the object is neces
sarily apprehended as solid.
This law of vision does not contradict that stated
above for distant objects : for when an object is removed
to a considerable distance, the images in the two eyes
coincide exactly, and the object is seen as single, though
without any direct apprehension of its solidity. The
first law is a special case of the second. Under the con
dition of exactly corresponding points, we have the per
ception of singleness, but no evidence of solidity. Under
the condition of nearly corresponding points, we may
have the perception of singleness, and with it, of solidity.
300 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
We have before noted it as an important feature in
our visual perception, that while we have two distinct
impressions upon the sense, which we can contemplate
separately and alternately, (the impressions on the two
eyes,) we have a higher perceptive faculty which can
recognize these two impressions, exactly similar to each
other, as only two images of one and the same assem
blage of objects. But we now see that the faculty by
which we perceive visible objects can do much more
than this : — it can not only unite two impressions, and
recognize them as belonging to one object in virtue of
their coincidence, but it can also unite and identify them,
even when they do not exactly coincide. It can correct
and adjust their small difference, so that they are both
apprehended as representations of the same figure. It
can infer from them a real form, not agreeing with
either of them ; and a solid space, which they are quite
incapable of exemplifying. The visual faculty decides
whether or not the two ocular images can be pictures of
the same solid object, and if they can, it undoubtingly
and necessarily accepts them as being so. This faculty
operates as if it had the power of calling before it all
possible solid figures, and of ascertaining by trial whether
any of those will, at the same time, fit both the outlines
which are given by the sense. It assumes the reality
of solid space, and, if it be possible, reconciles the appear
ances with that reality. And thus an activity of the
mind of a very remarkable and peculiar kind is exer
cised in the most common act of seeing.
10. It may be said that this doctrine, of such a visual
faculty as has been described, is very vague and obscure,
since we arc not told what are its limits. It adjusts and
corrects figures which nearly coincide, so as to identify
them. But lioir nearly, it may be asked, must the
figures approach oach other, in order that this adjust-
PECULIARITIES OF TJIE PERCEPTIONS. 301
inent may be possible * What discrepance renders im
possible the reconcilement of which we speak ? Is it
not impossible to give a definite answer to these ques
tions, and therefore impossible to lay down definitely
such laws of vision as we have stated ? To this I reply,
that the indefiniteness thus objected to us, is no new
difficulty, but one with which philosophers are familiar,
and to which they are already reconciled. It is, in fact,
no other than the indefiniteness of the limits of distinct
vision. How near to the face must an object be brought,
so that we shall cease to see it distinctly ? The distance,
it will be answered, is indefinite : it is different for
different persons; and for the same person, it varies
with the degree of effort, attention, and habit. But this
indefiniteness is only the indefiniteness, in another form,
of the deviation of the two ocular images from one
another : and in reply to the question concerning them
we must still say, as before, that in doubtful cases, the
power of apprehending an object as single, when this
can be done, will vary with effort, attention, and habit.
The assumption that the apparent object exists as a real
figure, in real space, is to be verified, if possible ; but,
in extreme cases, from the unfitness of the point of view,
or from any other cause of visual confusion or deception,
the existence of a real object corresponding to the ap
pearance may be doubtful ; as in any other kind of per
ception it may be doubtful whether our senses, under
disadvantageous circumstances, give us true information.
The vagueness of the limits, then, within which this
visual faculty can be successfully exercised, is no valid
argument against the existence of the faculty, or the
truth of the law which we have stated concerning its
action.
302 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
SECT. IV. — The Perception of Visible Figure.
11. Visible Figure. — There is one tenet on the
subject of vision which appears to me so extravagant
and unphilosophical, that I should not have thought it
necessary to notice it, if it had not been recently pro
mulgated by a writer of great acuteness in a book which
has obtained, for a metaphysical work, considerable cir
culation. I speak of Brown's opinion* that we have no
immediate perception of visible figure. I confess myself
unable to comprehend fully the doctrine which he would
substitute in the place of the one commonly received.
He states it thus t: "When the simple affection of sight
is blended with the ideas of suggestion [those arising
from touch, £c.] in what are termed the acquired per
ceptions of vision, as, for example, in the perception of
a sphere, it is colour only which is blended with the
large convexity, and not a small coloured plane." The
doctrine which Brown asserts in this and similar pas
sages, appears to be, that we do not by vision perceive
both colour and figure > ; but that the colour which we see
is blended with the figure which we learn the existence
of by other means, as by touch. But if this were pos
sible when we can call in other perceptions, how is it
possible when we cannot or do not touch the object ?
Why does the moon appear round, gibbous, or horned ?
What sense besides vision suggests to us the idea of her
figure ? And even in objects which we can reach, what
is that circumstance in the sense of vision which suggests
to us that the colour belongs to the sphere, except that
we see the colour where we see the sphere ? If we do
not see figure, we do not see position ; for figure is the
relative position of the parts of a boundary. If we do
not see position, why do we ascribe the yellow colour to
* Lectures, Vol. n. p. 82. + ll>. Vol. n. p. 90.
PECULIARITIES OF THE PERCEPTIONS. 303
the sphere on our left, rather than to the cube on our
right ? We associate the colour with the object, says
Dr. Brown ; but if his opinion were true, we could not
associate two colours with two objects, for we could
not apprehend the colours as occupying two different
places.
The whole .of Brown's reasoning on this subject is so
irreconcileable with the first facts of vision, that it is
difficult to conceive how it could proceed from a person
who has reasoned with great acuteness concerning touch.
In order to prove his assertion, he undertakes to ex
amine the only reasons which, he says*, he can imagine
for believing the immediate perception of visible figure :
(1) That it is absolutely impossible, in our present sen
sations of sight, to separate colour from extension ; and
(2) That there are, in fact, figures on the retina corre
sponding to the apparent figures of objects.
On the subject of the first reason, he says, that the
figure which we perceive as associated with colour, is the
real, and not the apparent figure. " Is there," he asks,
" the slightest consciousness of a perception of visible
figure, corresponding to the affected portion of the
retina?" To which, though he seems to think an affir
mative answer impossible, we cannot hesitate to reply,
that there is undoubtedly such a consciousness ; that
though obscured by being made the ground of habitual
inference as to the real figure, this consciousness is con
stantly referred to by the draughtsman, and easily re
called by any one. We may separate colour, he says
again f, from the figures on the retina, as we may sepa
rate it from length, breadth, and thickness, which we do
not see. But this is altogether false : we cannot separate
colour from length, breadth, and thickness, in any other
mty, than by transferring it to the visible figure which
* Lectures, Vol. n. p. $1. t //>. p. HJ.
304 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
we do see. He cannot, ho allows, separate the colour
from the visible form of the trunk of a large oak ; but
just as little, he thinks, can he separate it from the con
vex mass of the trunk, which (it is allowed on all hands)
he does not immediately see. But in this he is mis
taken : for if he were to make a picture of the oak, he
would separate the colour from the convex shape, which
he does not imitate, but he could 'not separate it from
the visible figure, which he does imitate ; and he would
then perceive that the fact that he has not an imme
diate perception of the convex form, is necessarily con
nected with the fact that he has an immediate percep
tion of the apparent figure ; so far is the rejection of
immediate perception in the former case from being a
reason for rejecting it in the latter.
Again, with regard to the second argument. It does
not, he says, follow, that because a certain figured por
tion of the retina is affected by light, we should see such
a figure ; for if a certain figured portion of the olfactory
organ were affected by odours, we should not acquire by
smell any perception of such figure *. This is merely to
say, that because we do not perceive position and figure
by one sense, we cannot do so by another. But this
again is altogether erroneous. It is an office of our
sight to inform us of position, and consequently of
figure ; for this purpose, the organ is so constructed
that the position of the object determines the position
of the point of the retina affected. There is nothing of
this kind in the organ of smell ; objects in different posi
tions and of different forms do not affect different parts
of the olfactory nerve, or portions of different shape.
Different objects, remote from each other, if perceived
by smell, affect the same part of the olfactory organs.
This is all quite intelligible; for it is not the office of
* Lecluiw, Vol. n. u. 8/.
PECULIARITIES OF THE PERCEPTIONS. 305
smell to inform us of position. Of what use or meaning
would be the curious and complex structure of the eye,
if it gave us only such vague and wandering notions of
the colours and forms of the flowers in a garden, as we
receive from their odours when we walk among them
blindfold? It is, as we have said, the prerogative of
vision to apprehend position : the places of objects on
the retina give this information. We do not suppose
that the affection of a certain shape of nervous expanse
will necessarily and in all cases give us the impression
of figure ; but we know that in vision it does ; and it is
clear that if we did not acquire our acquaintance with
visible figure in this way, we could not acquire it in
any way*.
The whole of this strange mistake of Brown's appears
to arise from the fault already noticed ; — that of consi
dering the image on the retina as the object instead of
the means of vision. This indeed is what he says : " the
true object of vision is not the distant body itself, but
the light that has reached the expansive termination of
the optic nerve f." Even if this were so, we do not see
why we should not perceive the position of the impres
sion on this expanded nerve. But as we have already
said, the impression on the nerve is the means of vision,
and enables us to assign a place, or at least a direction,
to the object from which the light proceeds, and thus
makes vision possible. Brown, indeed, pursues his own
peculiar view till he involves the subject in utter contu
sion. Thus he saysj, " According to the common theory
* When Brown says further (p. 87,) that we can indeed show the
image in the dissected eye ; but that " it is not in the dissected eye
that vision takes place;" it is difficult to see what his drift is. Does
he doubt that there is an imago formed in the living as completely as
in the dissected eye ?
* Lectures, Vol. 11. p. ;>7- + //>•, Vol. n. p. 89.
VOL. I. W. P. X
306 PIIILOSIMIY OF SECONDARY MECII ANICAL SCIENCES.
[that figure can be perceived by the eye,] a visible
sphere is at once to my perception convex and plane ;
and if the sphere be a large one, it is perceived at once
to be a sphere of many feet in diameter, and a plane
circular surface of the diameter of a quarter of an inch."
It is easy to deduce these and greater absurdities, if we
proceed on his strange and baseless supposition that the
object and the image on the retina are lot//, perceived.
But who is conscious of the image on the retina in anv
v
other way than as he sees the object by means of it?
Brown seems to have imagined that he was ana
lyzing the perception of figure in the same manner in
which Berkeley had analyzed the perception of distance.
lie ought to have recollected that such an undertaking,
to be successful, required him to show what elements he
analyzed it into. Berkeley analyzed the perception of
real figure into the interpretation of visible figure accord
ing to certain rules which he distinctly stated. Brown
analyzes the perception of visible figure into no ele
ments. Berkeley says, that we do not directly perceive
distance, but that we perceive something else, from
which we infer distance, namely, visible figure and colour,
and our own efforts in seeing; Brown says, that we do
not see figure, but infer it ; what then do we see, which
we infer it from? To this he offers no answer. He
asserts the seeming perception of visible figure to be a
result of " association ;" — of " suggestion." But what
meaning can we attach to this? Suggestion requires
something which suggests ; and not a hint is given what
it is which suggests position. Association implies two
things associated ; what is the sensation which we asso
ciate with form ? What is that visual perception which
is not figure, and which we mistake for figure ? What
perception is it that suggests a square to the eye ? What
impressions are those which have been associated with
PECULIAHITIKS OF THK 1'KKCKPTIONS. .'K)7
a visible triangle, so that the revival of the impressions
revives the notion of the triangle ? Brown has nowhere
pointed out such perceptions and impressions; nor indeed
was it possible for him to do so ; for the only visual
perceptions which he allows to remain, those of colour,
most assuredly do not suggest visible figures by their
differences; red is not associated with square rather than
with round, or with round rather than square. On the
contrary, the eye, constructed in a very complex and
wonderful manner in order that it may give to us directly
the perception of position as well as of colour, has it for
one of its prerogatives to give us this information ; and
the perception of the relative position of each part of
the visible boundary of an object constitutes the percep
tion of its apparent figure ; which faculty we cannot
deny to the eye without rejecting the plain and constant
evidence of our senses, making the mechanism of the
eye unmeaning, confounding the object with the means
of vision, and rendering the mental process of vision
utterly unintelligible.
Having sufficiently discussed the processes of per
ception, I now return to the consideration of the Ideas
which these processes assume.
CHAPTER III.
SUCCESSIVE ATTEMPTS AT THE SCIENTIFIC
APPLICATION OF THE IDEA OF A
MEDIUM.
1. IN what precedes, we have shown by various con
siderations that we necessarily and universally assume
the perception of secondary qualities to take place by
means of a medium interjacent between the object and
the person perceiving. Perception is affected by various
X2
308 PIIFLOSOI'HY OF SKCONDAKY M KC11 ANICAL SC1EXCKS.
peculiarities, according to the nature of the quality per
ceived : but in all cases a medium is equally essential to
the process.
This principle, which,, as we have seen, is accepted as
evident by the common understanding of mankind, is
confirmed by all additional reflection and discipline of
the mind, and is the foundation of all the theories which
have been proposed concerning the processes by which
the perception takes place, and concerning the modifi
cations of the qualities thus perceived. The medium, and
the mode in which the impression is conveyed through
the medium, seem to be different for different qualities ;
but the existence of the medium leads to certain neces
sary conditions or alternatives, which have successively
made their appearance in science, in the course of the
attempts of men to theorize concerning the principal
secondary qualities, sound, light, and heat. We must
now point out some of the ways, at first imperfect and
erroneous, in which the consequences of the fundamental
assumption were traced.
2. Sound. — In all cases the medium of sensation,
whatever it is, is supposed to produce the effect of con
veying secondary qualities to our perception by means
of its primary qualities. It was conceived to operate by
the size, form, and motion of its parts. This is a funda
mental principle of the class of sciences of which we
have at present to speak.
It was assumed from the first, as we have seen in the
passage lately quoted from Aristotle*, that in the con
veyance of sound, the medium of communication was
the air. But although the first theorists were right
so far, that circumstance did not prevent their going
entirely wrong when they had further to determine the
nature of the process. It was conceived by Aristotle
* Supr., p. 282.
SCIENTIFIC APPLICATION OF THE IDEA OF A MEDIUM. 300
that the air acted after the manner of a rigid body ;—
like a staff, which, receiving an impulse at one end, trans
mits it to the other. Now this is altogether an erro
neous view of the manner in which the air conveys the
impulse by which sound is perceived. An approach was
made to the true view of this process, by assimilating it
to the diffusion of the little circular waves which are
produced on the surface of still water when a stone is
dropt into it. These little waves begin from the point
thus disturbed, and run outwards, expanding on every
side, in concentric circles, till they are lost. The propa
gation of sound through the air from the point where it
is produced, was compared by Vitruvius to this diffu
sion of circular waves in water; and thus the notion of
a propagation of impulse by the waves of a fluid was
introduced, in the place of the former notion of the
impulse of an unyielding body.
But though, taking an enlarged view of the nature
of the progress of a wave, this is a just representation
of the motion of air in conveying sound, we cannot sup
pose that the process was, at the period of which we
speak, rightly understood. For the waves of water were
contemplated only as affecting the surface of the water ;
and as the air has no surface, the communication must
take place by means of an internal motion, which can
bear only a remote and obscure resemblance to the waves
which we see. And even with regard to the waves of
water, the mechanism by which they are produced and
transferred was not at all understood ; so that the com
parison employed by Vitruvius must be considered rather
as a loose analogy than as an exact scientific explanation.
No correct account of such motions was given, till
the formation of the science of Mechanics in modern
times had enabled philosophers to understand more dis
tinctly the mode in which motion is propagated through
310 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
a fluid, and to discern the forces which the process calls
into play, so as to continue the motion once begun.
Newton introduced into this subject the exact and rigor
ous conception of an undulation, which is the true key to
the explanation of impulses conveyed through a fluid.
Even at the present day. the right apprehension of
the nature of an undulation transmitted through a fluid
is found to be very difficult for all persons except those
whose minds have been duly disciplined by mathematical
studies. When we see a wave run along the surface of
water, we are apt to imagine at first that a portion of
the fluid is transferred bodily from one place to another.
But with a little consideration we may easily satisfy
ourselves that this is not so : for if we look at a field of
standing corn, when a breeze blows over it, we see waves
like those of water run along its surface. Yet it is clear
that in this case the separate stalks of corn only bend
backwards and forwards, and no portion of the grain is
really conveyed from one part of the field to the other.
This is obvious even to popular apprehension. The poet
speaks of
The rye,
That stoops its head when whirlwinds rave
And springs again in eddying wave
As each wild gust sweeps by.
Each particle of the mass in succession has a small
motion backwards and forwards ; and by this means a
large ridge made by many such particles runs along the
mass to any distance. This is the true conception of
an undulation in general.
Thus, when an undulation is propagated in a fluid,
it is not matter, but form, which is transmitted from one
place to another. The particles along the line of each
wave assume a certain arrangement, and this arrange
ment passes from one part to another, the particles
SCIENTIFIC APPLICATION OF THE IDEA OF A MEDIUM. 311
changing their places only within narrow limits, so as to
lend themselves successively to the arrangements by
which the successive waves, and the intervals between
the waves, are formed.
When such an undulation is propagated through
air, the wave is composed, not, as in water, of particles
which are higher than the rest, but of particles which
are closer to each other than the rest. The wave is not
a ridge of elevation, but a line of condensation ; and as
in water we have alternately elevated and depressed
lines, we have in air lines alternately condensed and
rarefied. And the motion of the particles is not, as in
water, up and down, in a direction transverse to that of
the wave which runs forwards ; in the motion of an
undulation •through air the motion of each particle is
alternately forwards and backwards, while the motion
of the undulation is constantly forwards.
This precise and detailed account of the undulatory
motion of air by which sound is transmitted was first
given by Newton. He further attempted to determine
the motions of the separate particles, and to point out
the force by which each particle affects the next, so as
to continue the progress of the undulation once begun.
The motions of each particle must be oscillatory; he
assumed the oscillations to be governed by the simplest
law of oscillation which had come under the notice of
mathematicians, (that of small vibrations of a pendulum;)
and he proved that in this manner the forces which are
called into play by the contraction and expansion of the
parts of the elastic fluid are such as the continuance of
the motion requires.
Newton's proof of the exact law of oscillatory motion
of the aerial particles was not considered satisfactory by
succeeding mathematicians ; for it was found that the
same result, the development of forces adequate to con-
312 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
tinue the motion, would follow if any other law of the
motion were assumed. Cramer proved this by a sort of
parody of Newton's proof, in which, by the alteration of
a few phrases in this formula of demonstration, it was
made to establish an entirely different conclusion.
But the general conception of an undulation as pre
sented by Newton was, as from its manifest mechanical
truth it could not fail to be, accepted by all mathemati
cians ; and in proportion as the methods of calculating
the motions of fluids were further improved, the neces
sary consequences of this conception, in the communica
tion of sound through air, were traced by unexceptionable
reasoning. This was especially done by Euler and
Lagrange, whose memoirs on such motions of fluids are
some of the most admirable examples which exist, of
refined mathematical methods applied to the solution of
difficult mechanical problems.
But the great step in the formation of the theory of
sound was undoubtedly that which we have noticed, the
introduction of the Conception of an Undulation such as
we have attempted to describe it: — a state, condition, or
arrangement of the particles of a fluid, which is trans
ferred from one part of space to another by means of
small motions of the particles, altogether distinct from
the movement of the undulation itself. This is a con
ception which is not obvious to common apprehension.
It appears paradoxical at first sight to speak of a large
wave (as the tide-wave) running up a river at the rate of
twenty miles an hour, while the stream of the river is
all the while flowing downwards. Yet this is a very
common fact. And the conception of such a motion
must be fully mastered by all who would reason rightly
* O «/
concerning the transmission of impressions through a
medium.
We have described the motion of sound as produced
SCIENTIFIC APPLICATION OF THE IDEA OF A MEDIUM. o!3
by small motions of the particle forwards and backwards,
while the waves, or condensed and rarefied lines, move
constantly forwards. It may be asked what right we
have to suppose the motion to be of this kind, since
when sound is heard, no such motions of the particles of
air can be observed, even by refined methods of observa
tion. Thus Bacon declares himself against the hypothesis
of such a vibration, since, as he remarks, it cannot be
perceived in any visible impression upon the flame of a
candle. And to this we reply, that the supposition of
this vibration is made in virtue of a principle which
is involved in the original assumption of a medium ;
namely, That a medium, in conveying secondary quali
ties, operates by means of its primary qualities, the
bulk, figure, motion, and other mechanical properties of
its parts. This is an Axiom belonging to the Idea of a
Medium. In virtue of this axiom it is demonstrable that
the motion of the air, when any how disturbed, must be
such as is supposed in our acoustical reasonings. For
the elasticity of the parts of the air, called into play by
its expansion and contraction, lead, by a mechanical
necessity, to such a motion as we have described. We
may add that, by proper contrivances, this motion may
be made perceptible in its visible effects. Thus the
theory of sound, as an impression conveyed through air,
is established upon evident general principles, although
the mathematical calculations which arc requisite to
investigate its consequences are, some of them, of a very
recondite kind.
'>. Liylit. — The early attempts to explain vision
represented it as performed by means of material rays
proceeding /Wftt the eye, by the help of which the eye
felt out the form and other visible qualities of an object,
as a blind man might do with his staff. lUit this opi
nion could not keep its ground long: for it did not even
HI4 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
explain the fact that light is necessary to vision. Light as
a peculiar medium was next assumed as the machinery
of vision ; but the mode in which the impression was
conveyed through the medium was left undetermined,
and no advance was made towards sound theory, on that
subject, by the ancients.
In modern times, when the prevalent philosophy
began to assume a mechanical turn (as in the theories
of Descartes), light was conceived to be a material sub
stance which is emitted from luminous bodies, and which
is also conveyed from all bodies to the eye, so as to
render them visible. The various changes of direction
by which the rays of light are affected, (reflexion, refrac
tion, &c.,) Descartes explained, by considering the par
ticles of light as small globules, which change their
direction when they impinge upon other bodies, accord
ing to the laws of mechanics. Newton, with a much
more profound knowledge of mechanics than Descartes
possessed, adopted, in the most mature of his specula
tions, nearly the same view of the nature of light ; and
endeavoured to show that reflexion, refraction, and other
properties of light, might be explained as the effects
which certain forces, emanating from the particles of
bodies, produce upon the luminiferous globules.
But though some of the properties of light could thus
be accounted for by the assumption of particles emitted
from luminous bodies, and reflected or refracted by forces,
other properties came into view which would not admit
of the same explanation. The phenomena of diffraction
(the fringes which accompany shadows) could never be
truly represented by such an hypothesis, in spite of many
attempts which were made. And the colours of thin
1>l«t(>s, which show the rays of light to be affected by an
alternation of two different conditions at small intervals
along their length, led Newton himself to incline, often
SCIENTIFIC APPLICATION OF THE IDEA OF A MEDIUM. 315
and strongly, to some hypothesis of undulation. The
double refraction of Iceland spar, a phenomenon in itself
very complex, could, it was found by Huyghcns, be
expressed with great simplicity by a certain hypothesis
of undulations.
Two hypotheses of the nature of the luminiferous
medium were thus brought under consideration ; the one
representing Light as Matter emitted from the luminous
object, the other, as Undulations propagated through a
fluid. These two hypotheses remained in presence of
each other during the whole of the last century, neither
of them gaining any material advantage over the other,
though the greater part of mathematicians, following
Newton, embraced the emission theory. But at the
beginning of the present century, an additional class of
phenomena, those of the interference of two rays of
light, were brought under consideration by Dr. Young;
and these phenomena were strongly in favour of the
undulatory theory, while they were irreconcilable with
the hypothesis of emission. If it had not been for the
original bias of Newton and his school to the other side,
there can be little doubt that from this period light as
well as sound would have been supposed to be pro
pagated by undulations; although in this case it was
necessary to assume as the vehicle of such undulations
a special medium or ether. Several points of the phe
nomena of vision no doubt remained unexplained by the
undulatory theory, as absorption, and the natural colours
of bodies ; but such facts, though they did not confirm,
did not evidently contradict the theory of a luminiferous
other ; and the facts which such a theory did explain, it
explained with singular happiness and accuracy.
But before this undulatory theory could begenerallv
accepted, it was presented in an entirely new point of
view by being combined with the facts of polarization.
31 G PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
The general idea of polarization must be illustrated here
after ; but we may here remark that Young and Fresnel,
who had adopted the undulatory theory, after being
embarrassed for some time by the new facts which were
thus presented to their notice, at last saw that these
facts might be explained by conceiving the vibrations to
be transverse to the ray, the motions of the particles
being not backwards and forwards in the line in which
the impulse travels, but to the right and left of that
line. This conception of transverse vibrations, though
quite unforeseen, had nothing in it which was at all diffi
cult to reconcile with the general notion of an undula
tion. We have described an undulation, or wave, as a
certain condition or arrangement of the particles of the
fluid successively transferred from one part of space to
another : and it is easily conceivable that this arrange
ment or wave may be -produced by a lateral transfer of
the particles from their quiescent positions. This con
ception of transverse vibrations being accepted, it was
found that the explanation of the phenomena of polari
zation and of those of interference led to the same
theory with a correspondence truly wonderful ; and this
coincidence in the views, collected from two quite dis
tinct classes of phenomena, was justly considered as an
almost demonstrative evidence of the truth of this undu-
latory theory.
It remained to be considered whether the doctrine
of transverse vibrations in a fluid could be reconciled
with the principles of mechanics. And it was found
that by making certain suppositions, in which no in
herent improbability existed, the hypothesis of trans
verse vibrations would explain the laws, both of inter
ference and of polarization of light, in air and in crystals
of all kinds, with a surprizing fertility and fidelity.
Thus the undulatory theory of light, like the undu-
SCIENTIFIC APPLICATION OF THE IDEA OF A MEDIUM. 317
latory theory of sound, is recommended by its conformity
to the fundamental principle of the Secondary Mecha
nical Sciences, that the medium must be supposed to
transmit its peculiar impulses according to the laws of
mechanics. Although no one had previously dreamt of
qualities being conveyed through a medium by such a
process, yet when it is once suggested as the only mode
of explaining some of the phenomena, there is nothing
to prevent our accepting it entirely, as a satisfactory
theory for all the known laws of light.
4. Heat. — With regard to heat as with regard to
light, a fluid medium was necessarily assumed as the
vehicle of the property. During the last century, this
medium was supposed to be an emitted fluid. And
many of the ascertained Laws of Heat, those which
prevail with regard to its radiation more especially, were
well explained by this hypothesis*. Other effects of
heat, however, as for instance latent heat^, and the
change of consistence of bodies^, were not satisfactorily
brought into connexion with the hypothesis1; while con
duction §, which at first did not appear to result from
the fundamental assumption, was to a certain extent
explained as internal radiation.
But it was by no means clear that an undulatory
theory of heat might not be made to explain these
phenomena equally well. Several philosophers inclined
to such a theory ; and finally, Ampere showed that the
doctrine that the heat of a body consists in the undula
tions of its particles propagated by means of the undula
tions of a medium, might be so adjusted as to explain all
which the theory of emission could explain, and more
over to account for facts and laws which were out of
* See the Account of the Theory of Exchanges, Hi.ff. Ind. Sci.,
B. x. c. i. sect. 2. t 76., c. ii. sect. 3.
£ 76., c. ii. sect. (2. § 7/>., c. i. sect. 7-
;{18 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
the reach of that theory. About the same time it was
discovered by Prof. Forbes and M. Nobili that radiant
heat is, under certain circumstances, polarized. Now
polarization had been most satisfactorily explained by
means of transverse undulations in the case of light ;
while all attempts to modify the emission theory so as to
include polarization in it, had been found ineffectual.
Hence this discovery was justly considered as lending
great countenance to the opinion that heat consists in
the vibrations of its proper medium.
But what is this medium ? Is it the same by which
the impressions of light are conveyed ? This is a difficult
question ; or rather it is one which we cannot at present
hope to answer with certainty. No doubt the con
nexion between light and heat is so intimate and con
stant, that we can hardly refrain from considering them
as affections of the same medium. But instead of
attempting to erect our systems on such loose and
general views of connexion, it is rather the business of
the philosophers of the present day to determine the
laws of the operation of heat, and its real relation to
light, in order that we may afterwards be able to con
nect the theories of the two qualities. Perhaps in a
more advanced state of our knowledge we may be able
to state it as an axiom, that two secondary qualities,
which are intimately connected in their causes and
effects, must be affections of the same medium. But at
present it does not appear safe to proceed upon such a
principle, although many writers, in their speculations
both concerning light and heat, and concerning other
properties, have not hesitated to do so.
Some other consequences follow from the Idea of a
Medium which must be the subject of another chapter.
CHAPTER IV.
OF THE MEASURE OF SECONDARY QUALITIES.
SECT. I. — Scales of Qualities in general.
THE ultimate object of our investigation in each of the
Secondary Mechanical Sciences, is the nature of the pro
cesses by which the special impressions of sound, light,
and heat, are conveyed, and the modifications of which
these processes are susceptible. And of this investiga
tion, as we have seen, the necessary basis is the principle,
that these impressions are transmitted by means of a
medium. But before we arrive at this ultimate object,
we may find it necessary to occupy ourselves with seve
ral intermediate objects : before we discover the cause,
it may be necessary to determine the lares of the phe
nomena. Even if we cannot immediately ascertain the
mechanism of light or heat, it may still be interesting
and important to arrange and measure the effects which
we observe.
The idea of a medium affects our proceeding in this
research also. We cannot measure secondary qualities
in the same manner in which we measure primary quali
ties, by a mere addition of parts. There is this leading
and remarkable difference, that while both classes of
qualities are susceptible of changes of magnitude, primary
qualities increase by addition of extension, secondary, by
augmentation of intensity. A space is doubled when
another equal space is placed by its side; one weight
joined to another makes up the sum of the two. But
when one degree of warmth is combined with another,
or one shade of red colour with another, we cannot in
like manner talk of the sum. The component parts do
not evidently retain their separate existence ; we cannot
:>'20 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
separate a strong green colour into two weaker ones, as
we can separate a large force into two smaller. The
increase is absorbed into the previous amount, and is no
longer in evidence as a part of the whole. And this is
the difference which has given birth to the two words
extended, and intense. That is extended which has
"partes extra paries," parts outside of parts: that is
intense which becomes stronger by some indirect and
unapparent increase of agency, like the stretching of the
internal springs of a machine, as the term intense im
plies. Extended magnitudes can at will be resolved
into the parts of which they were originally composed,
or any other which the nature of their extension admits ;
their proportion is apparent ; they are directly and at
once subject to the relations of number. Intensive
magnitudes cannot be resolved into smaller magnitudes ;
we can see that they differ, but we cannot tell in what
proportion ; we have no direct measure of their quan
tity. How many times hotter than blood is boiling-
water? The answer cannot be given by the aid of our
feelings of heat alone.
The difference, as we have said, is connected with
the fundamental principle that we do not perceive
secondary qualities directly, but through a medium. We
have no natural apprehension of light, or sound, or heat,
as they exist in the bodies from which they proceed, but
only as they affect our organs. We can only measure
them, therefore, by some Scale supplied by their effects.
And thus while extended magnitudes, as space, time, are
measurable directly and of themselves; intensive mag
nitudes, as brightness, loudness, heat, are measurable
only by artificial means and conventional scales. Space,
time, measure themselves : the repetition of a smaller
space, or time, while it composes a larger one, measures
it. But for light and heat we must have Photometers
MEASURE OF SECONDARY QUALITIES.
and Thermometers, which measure something which is
assumed to be an indication of the quality in question.
In one case, the mode of applying the measure, and
the meaning of the number resulting, are seen by intui
tion ; in the other, they are consequences of assumption
and reasoning. In the one case, they are Units, of
which the extension is made up ; in the other, they are
Degrees by which the intensity ascends.
2. When we discover any property in a sensible
quality, which at once refers us to number or space, we
readily take this property as a measure ; and thus we
make a transition from quality to quantity. Thus Pto
lemy in the third chapter of the First Book of his Har
monics begins thus : " As to the differences which exist
in sounds both in quality and in quantity, if we consider
that difference which refers to the acuteness and grave-
ness, we cannot at once tell to which of the above two
classes it belongs, till we have considered the causes of
such symptoms." But at the end of the chapter, having
satisfied himself that grave sounds result from the mag
nitude of the string or pipe, other things being equal,
he infers, " Thus the difference of acute and grave ap
pears to be a difference of quantity"
In the same manner, in order to form Secondary
Mechanical Sciences respecting any of the other pro
perties of bodies, we must reduce these properties to a
dependence upon quantity, and thus make them subject
to measurement. We cannot obtain any sciential truths
respecting the comparison of sensible qualities, till we
have discovered measures and scales of the qualities
which we have to consider; and accordingly, some of
the most important steps in such sciences have been the
establishment of such measures and scales, and the inven
tion of the requisite instruments.
The formation of the mathematical sciences which
VOL. i. w. P. Y
322 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
rest upon the measures of the intensity of sensible qua
lities took place mainly in the course of the last century.
Perhaps we may consider Lambert, a mathematician
who resided in Switzerland, and published about 1750,
as the person who first clearly felt the importance of
establishing- such sciences. His Photometry, Pyrometry,
HygTometry, are examples of the systematic reduction
of sensible qualities (light, heat, moisture) to modes of
numerical measurement.
We now proceed to speak of such modes of measure
ment with regard to the most obvious properties of
bodies.
SECT. II.— The Musical Scale.
3. THE establishment of the Harmonic Canon, that
is, of a Scale and Measure of the musical place of notes,
in the relation of high and low, was the first step in the
science of Harmonics. The perception of the differences
and relations of musical sounds is the office of the sense
of hearing ; but these relations are fixed, and rendered
accurately recognizable by artificial means. "Indeed,
in all the senses," as Ptolemy truly says in the opening
of his Harmonics, " the sense discovers what is approxi
mately true, and receives accuracy from another quarter:
the reason receives the approximately-true from another
quarter, and discovers the accurate truth." We can
have no measures of sensible qualities which do not
ultimately refer to the sense ; — whether they do this
immediately, as when we refer Colours to an assumed
Standard ; or mediately, as when we measure Heat by
Expansion, having previously found by an appeal to
sense that the expansion increases with the heat. Such
relations of sensible qualities cannot be described in
words, and can only be apprehended by their appropriate
faculty. The faculty by which the relations of sounds
MEASURE OF SECONDARY QUALITIES. 3*23
are apprehended is a musical ear in the largest accep
tation of the term. In this signification the faculty is
nearly universal among men; for all persons have musical
ears sufficiently delicate to understand and to imitate
the modulations corresponding to various emotions in
speaking; which modulations depend upon the succes
sion of acuter and graver tones. These are the relations
now spoken of, and these are plainly perceived by per
sons who have very imperfect musical ears, according to
the common use of the phrase. But the relations of
tones which occur in speaking are somewhat indefinite ;
and in forming that musical scale which is the basis of
our science upon the subject, we take the most definite
and marked of such relations of notes ; such as occur,
not in speaking but in singing. Those musical relations
of two sounds which we call the octare, the fftli, the
fourth, the third, are recognized after a short familiarity
with them. These chords or intervals are perceived to
have each a peculiar character, which separates them
from the relations of two sounds taken at random, and
makes it easy to know them when sung or played on
an instrument ; and for most persons, not difficult to
sing the sounds in succession exactly, or nearly correct.
These musical relations, or concords, then, are the ground
work of our musical standard. But how are we to name
these indescribable sensible characters? how to refer,
with unerring accuracy, to a type which exists only in
our own perceptions? We must have for this purpose
a Scale and a Standard.
The Musical Scale is a series of eight notes, ascend
ing by certain steps from the first or key-note to the
octave above it, each of the notes being fixed by such
distinguishable musical relations as we have spoken of
above. We may call these notes c, n, E, F, G, A, n, c ;
and we may then say that <; is determined by its being a
Y v
324 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
fifth above c ; D by its being a fourth below G ; E by its
being a third above c ; and similarly of the rest. It
will be recollected that the terms a fifth, a fourth, a
third, have hitherto been introduced as expressing cer
tain simple and indescribable musical relations among
sounds, which might have been indicated by any other
names. Thus we might call the fifth the dominant, and
the fourth the suldominant, as is done in one part of
musical science. But the names we have used, which
are the common ones, are in fact derived from the num
ber of notes which these intervals include in the scale
obtained in the above manner. The notes c, D, E, F, G,
being five, the interval from c to G is a fifth, and so of
the rest. The fixation of this scale gave the means of
describing exactly any note which occurs in the scale,
and the method is easily applicable to notes above and
below this range ; for in a series of sounds higher or
lower by an octave than this standard series, the ear
discovers a recurrence of the same relations so exact,
that a person may sometimes imagine he is producing
the same notes as another when he is singing the same
air an octave higher. Hence the next eight notes may
be conveniently denoted by a repetition of the same
letters, as the first ; thus, c, D, E, F, G, A, B, c, d, e, f, g,
a, b ; and it is easy to devise a continuation of such
cycles. And other admissible notes are designated by a
further modification of the standard ones, as by making
each note flat or sharp; which modification it is not
necessary here to consider, since our object is only to
show how a standard is attainable, and how it serves the
ends of science.
We may observe, however, that the above is not an
exact account of the first, or early Greek scale ; for this
scale was founded on a primary division of the interval
of two octaves (the extreme range which it admitted)
MEASURE OF SECONDARY QUALITIES. 325
into five tetrachords, each tetrachord including the in
terval of a fourth. All the notes of this series had
different names borrowed from this division* ; thus mese
was the middle or key-note; the note below it was
lichanos meson, the next below was parypate meson, the
next lower, hypate meson. The fifth above mese was
nete diazeugmendn, the octave was nete hyperbolcedn.
4. But supposing a complete system of such denomi
nations established, how could it be with certainty and
rigour applied ? The human ear is fallible, the organs
of voice imperfectly obedient; if this were not so, there
would be no such thing as a good ear or a good voice.
What means can be devised of finding at will a perfect
concord, a fifth or a fourth ? Or supposing such con
cords fixed by an acknowledged authority, how can they
be referred to, and the authority adduced? How can
we enact a Standard of sounds ?
A Standard was discovered in the Monochord. A
musical string properly stretched, may be made to pro
duce different notes, in proportion as we intercept a
longer or shorter portion, and make this portion vibrate.
The relation of the length of the strings which thus
sound the two notes G and c is fixed and constant, and
the same is true of all other notes. Hence the musical
interval of any notes of which we know the places in
the musical scale, may be reproduced by measuring the
lengths of string which are known to give them. If c
be of the length 180, D is 169, E is 144, F is 135, G is
1 20 ; and thus the musical relations are reduced to
numerical relations, and the monochord is a complete
and perfect Tonometer.
We have here taken the length of the string as the
measure of the tone : but we may observe that there is
in us a necessary tendency to assume that the ground
* Burney's History of Music, Vol. r. p. 28.
326 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
of this measure is to be sought in some ulterior cause ;
and when we consider the matter further, we find this
cause in the frequency of these vibrations of the string.
The truth that the same note must result from the same
frequency of vibration is readily assented to on a slight
suggestion of experience. Thus Mersenne*, when he
undertakes to determine the frequency of vibrations of a
given sound, says "Supponendum est quoscunque nervos
et quaslibet chordas unisonum facientes eundem efficere
numerum rccursuum codem vel equali tempore, quod
perpetua constat experientia." And he proceeds to
apply it to cases where experience could not verify this
assertion, or at least had not verified it, as to that of
pipes.
The pursuit of these numerical relations of tones
forms the science of Harmonics ; of which here we do
not pretend to give an account, but only to show, how
the invention of a Scale and Nomenclature, a Standard
and Measure of the tone of sounds, is its necessary basis.
We will therefore now proceed to speak of another sub
ject; colour.
SECT. III. — Scales of Colour.
5. The Prismatic Scale of Colour. — A SCALE of
Colour must depend originally upon differences discern
ible by the eye, as a scale of notes depends on differences
perceived by the ear. In one respect the difficulty is
greater in the case of the visible qualities, for there are
no relations of colour which the eye peculiarly singles
out and distinguishes, as the ear selects and distinguishes
an octave or a fifth. Hence we are compelled to take
an arbitrary scale; and we have to find one which is
fixed, and which includes a proper collection of colours.
The prismatic spectrum, or coloured image produced
* Hannonia, Lib. 11. Prop. 19.
MEASURE OF SECONDARY QUALITIES. 327
when a small beam of light passes obliquely through
uny transparent surface (as the surface of a prism of
glass,) otters an obvious Standard as far as it is appli
cable. Accordingly colours have, for various purposes,
been designated by their place in the spectrum ever
since the time of Newton ; and we have thus a means of
referring to such colours as are included in the series
red, orange, yellow, green, blue, violet, indigo, and the
intermediate tints.
But this scale is not capable of numerical precision.
If the spectrum could be exactly defined as to its ex
tremities, and if these colours occupied always the same
proportional part of it, we might describe any colour in
the above series by the measure of its position. But
the fact is otherwise. The spectrum is too indefinite in
its boundaries to afford any distinct point from which
we may commence our measures ; and moreover the
spectra produced by different transparent bodies differ
from each other. Newton had supposed that the spec
trum and its parts wrere the same, so long as the refrac
tion was the same ; but his successors discovered that,
with the same amount of refraction in different kinds of
glass, there are different magnitudes of the spectrum ;
and what is still worse with reference to our present
purpose, that the spectra from different glasses have
the colours distributed in different proportions. In order,
therefore, to make the spectrum the scale of colour, we
must assume some fixed substance ; for instance, wre may
take water, and thus a series approaching to the colours
of the rainbow will be our standard. But we should
still have an extreme difficulty in applying such a rule.
The distinctions of colour which the terms of common
language express, are not used with perfect unanimity
or with rigorous precision. What one person calls bluish
green another calls greenish blue. Nobody can say
328 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
what is the precise boundary between red and orange.
Thus the prismatic scale of colour was incapable of
mathematical exactness, and this inconvenience was felt
up to our own times.
But this difficulty was removed by a curious dis
covery of Wollaston and Fraunhofer ; who found that
there are, in the solar spectrum, certain fine black Lines
which occupy a definite place in the series of colours,
and can be observed with perfect precision. We have
now no uncertainty as to what coloured light we are
speaking of, when we describe it as that part of the
spectrum in which Fraunhofer's Line c or D occurs.
And thus, by this discovery, the prismatic spectrum of
sunlight became, for certain purposes, an exact Chroma-
tomefer.
6. Neifton's Scale of Colours. — Still, such a standard,
though definite, is arbitrary and seemingly anomalous.
The lines A, B, c, D, &c., of Fraunhofer's spectrum are
distributed without any apparent order or law ; and we
do not, in this way, obtain numerical measures, which is
what, in all cases, we desire to have. Another discovery
of Newton, however, gives us a spectrum containing the
same colours as the prismatic spectrum, but produced in
another way, so that the colours have a numerical rela
tion. I speak of the laws of the colours of thin plates.
The little rainbows which we sometimes see in the cracks
of broken glass are governed by fixed and simple laws.
The kind of colour produced at any point depends on
the thickness of the thin plate of air included in the fis
sure. If the thickness be eight-millionths of an inch,
the colour is orange, if fifteen-millionths of an inch, we
have green, and so on ; and thus these numbers which
succeed each other in a regular order from red to indigo,
give a numerical measure of each colour ; which mea
sure, when we pursue the subject, we find is one of the
MEASURE OF SECONDARY QUALITIES.
bases of all optical theory. The series of colours ob
tained from plates of air of gradually increasing thick
ness is called Newton s Scale of Colours ; but we may
observe that this is not precisely what we are here speak
ing of, a scale of simple colours ; it is a series produced
by certain combinations, resulting from the repetition of
the first spectrum, and is mainly useful as a standard for
similar phenomena, and not for colour in general. The
real scale of colour is to be found, as we have said, in
the numbers which express the thickness of the pro
ducing film; — in the length of a Jit in Newton's phrase
ology, or the length of an undulation in the modern
theory.
7. Scales of Impure Colours. — The standards just
spoken of include (mainly at least) only pure and simple
colours ; and however complete they may be for certain
objects of the science of optics, they are insufficient for
other purposes. They do not enable us to put in their
place mixed and impure colours. And there is, in the
case of colour, a difficulty already noticed, which does
not occur in the case of sound ; two notes, when sounded
together, are not necessarily heard as one; they are
recognized as still two, and as forming a concord or a
discord. But two colours form a single colour ; and the
eye cannot, in any way, distinguish between a green
compounded of blue and yellow, and the simple, unde-
composable green of the spectrum. By composition of
three or more colours, innumerable new colours may be
generated which form no part of the prismatic series ;
and by such compositions is w.oven the infinitely varied
web of colour which forms the clothing of nature. How
are we to classify and arrange all the possible colours
of objects, so that each shall have a place and name?
How shall we find a chromato meter for impure as well
as for pure colour ?
330 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
Though no optical investigations have depended on a
scale of impure colours, such a scale has been wanted
and invented for other purposes ; for instance, in order
to identity and describe objects of natural history. Not
to speak of earlier essays, we may notice Werner's No
menclature of Colours, devised for the purpose of de
scribing minerals. This scale of colour was far superior
to any which had previously been promulgated. It was,
indeed, arbitrary in the selection of its degrees, and in
a great measure in their arrangement ; and the colours
were described by the usual terms, though generally
with some added distinction ; as blackish green, bluish
green, apple-green, emerald-green. But the great merit
of the scale was its giving a fixed conventional meaning
to these terms, so that they lost much of their usual
vagueness. Thus apple-green did not mean the colour
of any green apple casually taken ; but a certain definite
colour which the student was to bear in mind, whether or
not he had ever seen an apple of that exact hue. The
words were not a description, but a record of the colour :
the memory was to retain a sensation, not a name.
The imperfection of the system (arising from its ar
bitrary form) was its incompleteness : however well it
served for the reference of the colours which it did con
tain, it was applicable to no others; and thus, though
Werner's enumeration extended to more than a hundred
colours, there occur in nature a still greater number
which cannot be exactly described by means of it.
In such cases the imclassed colour is, by the Werne-
rians, defined by stating it as intermediate between two
others : thus we have an object described as between
emerald-green and grass-green. The eye is capable of
perceiving a gradation from one colour to another ; such
as may be produced by a gradual mixture in various
ways. And if we image to ourselves such a mixture, we
MEASURE OF SECONDARY QUALITIES. 331
can compare with it a given colour. But in employing
this method we have nothing to tell us in what part of
the scale we must seek for an approximation to our un-
classed colour. We have no rule for discovering where
we are to look for the boundaries of the definition of a
colour which the Wernerian series does not supply.
For it is not always between contiguous members of the
series that the undescribed colour is found. If we place
emerald-green between apple-green and grass-green, we
may yet have a colour intermediate between emerald-
green and leek-green ; and, in fact, the Wernerian series
of colours is destitute of a principle of self-arrangement
and gradation ; and is thus necessarily and incurably
imperfect.
8. We should have a complete Scale of Colours, if
we could form a series including all colours, and arranged
so that each colour was intermediate in its tint between
the adjacent terms of the series ; for then, whether we
took many or few of the steps of the series for our
standard terms, the rest could be supplied by the law of
continuity; and any given colour would either cor
respond to one of the steps of our scale or fall between
two intermediate ones. The invention of a Chroma-
, tometer for Impure Colours, therefore, requires that we
should be able to form all possible colours by such inter
mediation in a systematic manner ; that is, by the mix
ture or combination of certain elementary colours ac
cording to a simple rule : and we are led to ask whether
such a process has been shown to be possible.
The colours of the prismatic spectrum obviously do
form a continuous series ; green is intermediate between
its neighbours yellow and blue, orange between red and
yellow ; and if we suppose the two ends of the spectrum
bent round to meet each other, so that the arrangement
of the colours may be circular, the violet and indigo will
332 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
find their appropriate place between the blue and red.
And all the interjacent tints of the spectrum, as well as
the ones thus named, will result from such an arrange
ment. Thus all the pure colours are produced by com
binations two and two of three primary colours, red,
yellow, and blue ; and the question suggests itself
whether these three are not really the only primary
colours, and whether all the impure colours do not arise
from mixtures of the three in various proportions.
There are various modes in which this suggestion may
be applied to the construction of a scale of colours ; but
the simplest, and the one which appears really to verify
the conjecture that all possible colours may be so ex
hibited, is the following. A certain combination of red,
yellow, and blue, will produce black, or pure grey, and
when diluted, will give all the shades of grey which
intervene between black and white. By adding various
shades of grey, then, to pure colours, we may obtain all
the possible ternary combinations of red, yellow, and
blue ; and in this way it is found that we exhaust the
range of colours. Thus the circle of pure colours of
which we have spoken may be accompanied by several
other circles, in which these colours are tinged with a
less or greater shade of grey ; and in this manner it is
found that we have a perfect chromatometer ; every
possible colour being exhibited either exactly or by
means of approximate and contiguous limits. The ar
rangement of colours has been brought into this final
and complete form by M. Merimee, whose Chromatic
Scale is published by M. Mirbel in his Elements of Bo
tany. We may observe that such a standard affords us
a numerical exponent for every colour by means of the
proportions of the three primary colours which compose
it ; or, expressing the same result otherwise, by means
of the pure colour which is involved, and the proportion
MEASURE OF SECONDARY QUALITIES. 333
of grey by which it is rendered impure. In such a
scale the fundamental elements would be the precise
tints of red, yellow, and blue which are found or as
sumed to be primary ; the numerical exponents of each
colour would depend upon the arbitrary number of de
grees which we interpose between each two primary
colours; and between each pure colour and absolute
blackness. No such numerical scale has, however, as yet,
obtained general acceptation'-.
SECT. IV. — Scales of Light.
9. Photometer. — ANOTHER instrument much needed
in optical researches is a Photometer, a measure of the
intensity of light. In this case, also, the organ of sense,
the eye, is the ultimate judge ; nor has any effect of
light, as light, yet been discovered which we can sub
stitute for such a judgment. All instruments, such as
that of Leslie, which employ the heating effect of light,
or at least all that have hitherto been proposed, are in
admissible as photometers. But though the eye can
* The reference to Fraunhofer's Lines, as a means of determining
the place of a colour in the prismatic series, has been objected to,
because, as is asserted, the colours which are in the neighbourhood of
each line vary with the position of the sun, state of the atmosphere
and the like. It is very evident that coloured light refracted by the
prism will not give the same spectrum as white light. The spectrum
given by white light is of course the one here meant. It is an usual
practice of optical experimenters to refer to the colours of such a
spectrum, defining them by Fraunhofer's Lines.
I do not know whether it needs explanation that the " first spec
trum" in Newton's rings is a ring of the prismatic colours.
I have not had an opportunity of consulting Lambert's Photometria,
sive de mensura et gradibus luminis, colorum, el umbra;, published in
17<>0, nor Mayer's Commentatio de Affinitate Colorum, (1758,) in
which, I believe, he describes a chromatometer. The present work is
not intended to be complete as a history ; and I hope I have given
sufficient historical detail to answer its philosophical purpose.
334 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
judge of two surfaces illuminated by light of the same
colour, and can determine when they are equally bright,
or which is the brighter, the eye can by no means decide
at sight the proportion of illumination. How much in
such judgments we are affected by contrast, is easily seen
when we consider how different is the apparent bright
ness of the moon at mid-day and at midnight, though
the light which we receive from her is, in fact, the same
at both periods. In order to apply a scale in this case,
we must take advantage of the known numerical rela
tions of lidit. We are certain that if all other illumi-
o
nation be excluded, two equal luminaries, under the
same circumstances, will produce an illumination twice
as great as one does ; and we can easily prove, from ma
thematical considerations, that if light be not enfeebled
by the medium through which it passes, the illumination
on a given surface will diminish as the square of the
distance of the luminary increases. If, therefore, we
can by taking a fraction thus known of the illuminating
effect of one luminary, make it equal to the total effect
of another, of which equality the eye is a competent
judge, we compare the effects of the two luminaries. In
order to make this comparison we may, with Rumford,
look at the shadows of the same object made by the two
lights, or with Ritchie, we may view the brightness pro
duced on two contiguous surfaces, framing an apparatus
so that the equality may be brought about by proper
adjustment ; and thus a measure will become practica
ble. Or we may employ other methods as was done by
Wollaston "-, who reduced the light of the sun by observ
ing it as reflected from a bright globule, and thus found
the light of the sun to be 10,000,000,000 times that of
Sirius, the brightest fixed star. All these methods are
inaccurate, even as methods of comparison ; and do not
* Phil. Trans., 1829, p. 19.
MEASURE OF SECONDARY QUALITIES. 33 ~>
offer any fixed or convenient numerical standard ; but
none better have yet been devised"".
10. Cyanometer. — As we thus measure the brightness
of a colourless light, we may measure the intensity of
any particular colour in the same way; that is, by apply
ing a standard exhibiting the gradations of the colour in
question till we find a shade which is seen to agree with
the proposed object. Such an instrument we have in
the Cyanometer, which was invented by Saussure for the
purpose of measuring the intensity of the blue colour of
the sky. We may introduce into such an instrument a
numerical scale, but the numbers in such a scale will be
altogether arbitrary.
SECT. V. — Scales of Heat.
11. Thermometers. — WHEN we proceed to the sensa
tion of heat, and seek a measure of that quality, we find,
at first sight, new difficulties. Our sensations of this
kind are more fluctuating than those of vision ; for we
know that the same object may feel warm to one hand
and cold to another at the same instant, if the hands
have been previously cooled and warmed respectively.
Nor can we obtain here, as in the case of light, self-evi
dent numerical relations of the heat communicated in
given circumstances ; for we know that the effect so pro
duced will depend on the warmth of the body to be
heated, as well as on that of the source of heat ; the
summer sun, which warms our bodies, will not augment
the heat of a red-hot iron. The cause of the differ
ence of these cases is, that bodies do not receive the
whole of their heat, as they receive the whole of their
light, from the immediate influence of obvious external
* Improved Photometers have been devised by Professor Wheat-
stone, Professor Potter, and Professor Steinheil ; but they depend upon
principles similar to those mentioned in the text.
336 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
agents. There is no readily-discovered absolute cold,
corresponding to the absolute darkness which we can
easily produce or imagine. Hence we should be greatly
at a loss to devise a Thermometer, if we did not find an
indirect effect of heat sufficiently constant and measurable
to answer this purpose. We discover, however, such an
effect in the expansion of bodies by the effect of heat.
12. Many obvious phenomena show that air, under
given circumstances, expands by the effect of heat ; the
same is seen to be true of liquids, as of water, and spirit
of wine ; and the property is found to belong also to the
metallic fluid, quicksilver. A more careful examination
showed that the increase of bulk in some of these bodies
by increase of heat was a fact of a nature sufficiently
constant and regular to afford a means of measuring that
previously intangible quality ; and the Thermometer was
invented. There were, however, many difficulties to
overcome, and many points to settle, before this instru
ment was fit for the purposes of science.
An explanation of the way in which this was done
necessarily includes an important chapter of the history
of Thermotics. We must now, therefore, briefly notice
historically the progress of the Thermometer. The lead
ing steps of this progress, after the first invention of the
instrument, were — The establishment of fixed points in
the thermometric scale — The comparison of the scales
of different substances — And the reconcilement of these
differences by some method of interpreting them as indi
cations of the absolute quantity of heat.
13. It would occupy too much space to give in detail
the history of the successive attempts by which these
steps were effected. A thermometer is described by
Bacon under the title Vitrum Calendare ; this was an
air thermometer. Newton used a thermometer of linseed
oil, and he perceived that the first step requisite to give
MEASURE OF SECONDARY QUALITIES. 337
value to such an instrument was to fix its scale ; accord
ingly he proposed his Scala Graduum Caloris*. But
when thermometers of different liquids were compared,
it appeared, from their discrepancies, that this fixation
of the scale of heat was more difficult than had been
supposed. It was, however, effected. Newton had taken
freezing water, or rather thawing snow, as the zero of
his scale, which is really a fixed point; Halley and Amon-
tons discovered (in 1693 and 1702) that the heat of
boiling water is another fixed point ; and Daniel Gabriel
Fahrenheit, of Dantzig, by carefully applying these two
standard points, produced, about 1714, thermometers,
which were constantly consistent with each other. This
result was much admired at the time, and was, in fact,
the solution of the problem just stated, the fixation of
the scale of heat.
14. But the scale thus obtained is a conventional
not a natural scale. It depends upon the fluid employed
for the thermometer. The progress of expansion from
the heat of freezing to that of boiling water is different
for mercury, oil, water, spirit of wine, air. A degree of
heat which is half-way between these two standard
points according to a mercurial thermometer, will be .
below the half-way point in a spirit thermometer, and
above it in an air thermometer. Each liquid has its
own march in the course of its expansion. Deluc and
others compared the marches of various liquids, and
thus made what we may call a concordance of thermo
meters of various kinds.
15. Here the question further occurs : Is there not
some natural measure of the degrees of heat ? It ap
pears certain that there must be such a measure, and
that by means of it all the scales of different liquids
must be reconciled. Yet this does not seem to have
* Phil. Trans., 1701.
VOL. I. W. P. /
338 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
occurred at once to men's minds. Deluc, in speaking
of the researches which we have just mentioned, says*,
"When I undertook these experiments, it never once
came into my thoughts that they could conduct me with
any probability to a table of real degrees of heat. But
hope grows with success, and desire with hope." Accord
ingly he pursued this inquiry for a long course of years.
What are the principles by which we are to be
guided to the true measure of heat ? Here, as in all the
sciences of this class, we have the general principle, that
the secondary quality, heat, must be supposed to be per
ceived in some way by a material medium or fluid. If
we take that which is, perhaps, the simplest form of this
hypothesis, that the heat depends upon the quantity of
this fluid, or caloric, which is present, we shall find that
we are led to propositions which may serve as a foun
dation for a natural measure of heat. The Method of
Mixtures is one example of such a result. If we mix
together two pints of water, one hot and one cold, is it
not manifest that the temperature of the mixture must
be midway between the two? Each of the two portions
brings with it its own heat. The whole heat, or caloric,
of the mixture is the sum of the two ; and the heat of
each half must be the half of this sum, and therefore its
temperature must be intermediate between the tempe
ratures of the equal portions which were mixed. Deluc
made experiments founded upon this principle, and was
led by them to conclude that "the dilatations of mer
cury follow an accelerated march for successive equal
augmentations of heat."
But there are various circumstances which prevent
this method of mixtures from being so satisfactory as
at first sight it seems to promise to be. The different
capacities for heat of different substances, and even of
* Modif. tie 1'Ahnosph., 1782, p. 303.
MEASURE OF SECONDARY QUALITIES. 339
the same substance at different temperatures, introduce
much difficulty into the experiments; and this path of
inquiry has not yet led to a satisfactory result.
16. Another mode of inquiring into the natural
measure of heat is to seek it by researches on the laic
of cooling of hot bodies. If we assume that the process
of cooling of hot bodies consists in a certain material
heat flying off, we may, by means of certain probable
hypotheses, determine mathematically the law according
to which the temperature decreases as time goes on ; and
we may assume that to be the true measure of tempe
rature which gives to the experimental law of cooling
the most simple and probable form.
It appears evident from the most obvious conceptions
which we can form of the manner in which a body parts
with its superabundant heat, that the hotter a body is,
the faster it cools ; though it is not clear without expe
riment, by what law the rate of cooling will depend upon
the heat of the body. Newton took for granted the
most simple and seemingly natural law of this depend
ence : he supposed the rate of cooling to be proportional
to the temperature, and from this supposition he could
deduce the temperature of a hot iron, calculating from
the original temperature and the time during which it
had been cooling. By calculation founded on such a
basis, he graduated his thermometer.
17. But a little further consideration showed tli.it
the rate of cooling of hot bodies depended upon the
temperature of the surrounding bodies, as well as upon
its own temperature. Prevost's Theory of Exchanges*
was propounded with a view of explaining this depend
ence, and was generally accepted. According to this
theory, all bodies radiate heat to one another, and are
thus constantly giving and receiving heat; and a body
* Pecherches stir In Cha/cur, 1791- Hi»t. Ind. Sci., B. x. c. i. sect. 2.
340 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
which is hotter than surrounding bodies, cools itself,
and warms the surrounding bodies, by an exchange of
heat for heat, in which they are the gainers. Hence if
6 be the temperature of the bodies, or of the space, by
which the hot body is surrounded, and 0 + t the tempe
rature of the hot body, the rate of cooling will depend
upon the excess of the radiation for a temperature 6> + t,
above the radiation for a temperature 0.
Accordingly, in the admirable researches of MM.
Dulong and Petit upon the cooling of bodies, it was
assumed that the rate of cooling of the hot body was
represented by the excess of F (0 + t) above F (0); where
F represented some mathematical function, that is, some
expression obtained by arithmetical operations from the
temperatures 9 + t and 0: although what these operations
are to be, was left undecided, and was in fact determined
by the experiments. And the result of their investiga
tions was, that the function is of this kind : — when -the
temperature increases by equal intervals, the function
increases in a continued geometric proportion""". This
was, in fact, the same law which had been assumed by
Newton and others, with this difference, that they had
neglected the term which depends upon the temperature
of the surrounding space.
18. This law falls in so well with the best concep
tions we can form of the mechanism of cooling upon the
supposition of a radiant fluid caloric, that it gives great
probability to the scale of temperature on which the
simplicity of the result depends. Now the temperatures
in the formula? just referred to were expressed by means
of the air thermometer. Hence MM. Dulong and Petit
justly state that while all different substances employed
" The formula for the rate of cooling is mae+t — ma0. where the
quantity m depends upon the nature of the body, the state of its sur
face, and other circumstances. — Ann. Chim. vn. loO.
MEASURE OF SECONDARY QUALITIES. :J41
as thermometers give different laws of thermotical phe
nomena, their own success in obtaining simple and
general laws by means of the air thermometer, is a strong
recommendation of that as the natural scale of heat.
They add*, " The well-known uniformity of the principal
physical properties of all gases, and especially the per
fect identity of their laws of dilatation by heat, [a very
important discovery of Dalton and Gay Lussacf,] make
it very probable that in this class of bodies the disturb
ing causes have not the same influence as in solids and
liquids ; and consequently that the changes of bulk pro
duced by the action of heat are here in a more imme
diate dependence on the force which produces them."
19. Still we cannot consider this point as settled
till we obtain a more complete theoretical insight into
the nature of heat itself. If it be true that heat con
sists in the vibrations of a fluid, then, although, as
Ampere has shown |, the laws of radiation will, on
mathematical grounds, be the same as they are on the
hypothesis of emission, we cannot consider the natural
scale of heat as determined, till we have discovered some
means of measuring the caloriferous vibrations as we
measure luminiferous vibrations. We shall only know
what the quantity of heat is when we know what heat
itself is ; — when we have obtained a theory which satis
factorily explains the manner in which the substance or
medium of heat produces it effects. When we see how
radiation and conduction, dilatation and liquefaction, are
all produced by mechanical changes of the same fluid,
we shall then see what the nature of that change is
which dilatation really measures, and what relation it
bears to any more proper standard of heat.
We may add, that while our thermotical theory is
* Annalcsde Chimie, vn. lf>3. t Hist. Ind. Sci., B. x. c. ii. sect. 1.
I //>.. c. iv.
34*2 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
still so imperfect as it is, all attempts to divine the true
nature of the relation between light and heat are pre
mature, and must be in the highest degree insecure
and visionary. Speculations in which, from the general
assumption of a caloriferous and luminiferous medium,
and from a few facts arbitrarily selected and loosely
analyzed, a general theory of light and heat is asserted,
are entirely foreign to the course of inductive science,
and cannot lead to any stable and substantial truth.
•JO. Other Instruments for measuring Heat. — It
does not belong to our present purpose to speak of
instruments of which the object is to measure, not sen
sible qualities, but some effect or modification of the
cause by which such qualities are produced : such, for
instance, are the Calorimeter, employed by Lavoisier
and Laplace, in order to compare the specific lieat of
different substances ; and the Actiiwmeter, invented by
Sir John Herschel, in order to determine the effect of
the suns rai/s by means of the heat which they commu
nicate in a given time ; which effect is, as may readily
be supposed, very different under different circumstances
of atmosphere and position. The laws of such effects
may be valuable contributions to our knowledge of heat,
but the interpretation of them must depend on a pre
vious knowledge of the relations which temperature bears
to heat, according to the views just explained.
SECT. VI. — Scales of other Qualities.
21. BEFORE quitting the subject of the measures of
sensible qualities, we may observe that there are several
other such qualities for which it would be necessary to
have scales and means of measuring, in order to make
any approach to science on such subjects. This is true,
for instance, of tastes and smells. Indeed some attempts
have been made towards a classification of the tastes of
MEASURE OF SECONDARY QUALITIES. 343
sapid substances, but these have not yet assumed any
satisfactory or systematic character ; and I am not aware
that any instruments has been suggested for measuring
either the flavour or the odour of bodies which possess
such qualities.
22. Quality of Sounds. — The same is true of that
kind of difference in sounds which is peculiarly termed
their quality ; that character by which, for instance, the
sound of a flute differs from that of a hautbois, when the
note is the same ; or a woman's voice from a boy's.
23. Articulate Sounds. — There is also in sounds
another difference, of which the nature is still obscure,
but in reducing which to rule, and consequently to mea
sure, some progress has nevertheless been made. I
speak of the differences of sound considered as articulate.
Classifications of the sounds of the usual alphabets have
been frequently proposed ; for instance, that which ar
ranges the consonants in the following groups :
Sharp. Flat. Sharp Aspirate1. Flat Aspirate. Nasal.
p b ph ( /) bh (i>) m
k g (hard) kh gh ng
t (\ th (sharp) th (flat) n
s x sh zh
It is easily perceived that the relations of the sounds in
each of these horizontal lines are analogous ; and accord
ingly the rules of derivation and modification of words
in several languages proceed upon such analogies. In
the same manner the vowels may be arranged in an order
depending on their sound. But to make such arrange
ments fixed and indisputable, we ought to know the
mechanism by which such modifications are caused. In
struments have been invented by which some of these
sounds can be imitated ; and if such instruments could
be made to produce the above series of articulate sounds,
by connected and regular processes, we should find, in
.'344 PHILOSOPHY OF SECONDARY MECHANICAL SCIENCES.
the process, a measure of the sound produced. This
has been in a great degree effected for the Vowels by
Professor Willis's artificial mode of imitating them. For
he finds that if a musical reed be made to sound through
a cylindrical pipe, we obtain by gradually lengthening
the cylindrical pipe, the series of vowels I, E, A, o, u,
with intermediate sounds""". In this instrument, then,
the length of the pipe would determine the vowel, and
might be used numerically to express it. Such an in
strument so employed would be a measure of vowel
quality, and might be called a Phthongometer.
Our business at present, however, is not with instru
ments which might be devised for measuring sensible
qualities, but with those which have been so used, and
have thus been the basis of the sciences in which such
qualities are treated of; and this we have now done suf
ficiently for our present purpose.
24. There is another Idea which, though hitherto
very vaguely entertained, has had considerable influence
in the formation, both of the sciences spoken of in the
present Book, and on others which will hereafter come
under our notice : namely, the Idea of Polarity. This
Idea will be the subject of the ensuing Book. And
although this Idea forms a part of the basis of various
other extensive portions of science, as Optics and Che
mistry, it occupies so peculiarly conspicuous a place in
speculations belonging to what I have termed the Mecha-
nico-Chemical Sciences, (Magnetism and Electricity,)
that I shall designate the discussion of the Idea of
Polarity as the Philosophy of those Sciences.
* Camb. Trans., A"ol. in. p. 239.
345
BOOK V.
OF THE PHILOSOPHY OF THE MECHANICO-
CHEMICAL SCIENCES.
CHAPTER I.
ATTEMPTS AT THE SCIENTIFIC APPLICATION
OF THE IDEA OF POLARITY.
1. IN some of the mechanical sciences, as Magnetism
and Optics, the phenomena are found to depend upon
position (the position of the magnet, or of the ray of
light,) in a peculiar alternate manner. This dependence,
as it was first apprehended, was represented by means
of certain conceptions of space and force, as for instance
by considering the two poles of a magnet. But in all
such modes of representing these alternations by the
conceptions borrowed from other ideas, a closer exami
nation detected something superfluous and something
defective ; and in proportion as the view which philo
sophers took of this relation was gradually purified from
these incongruous elements, and was rendered more
general and abstract by the discovery of analogous pro
perties in new cases, it was perceived that the relation
could not be adequately apprehended without consider
ing it as involving a peculiar and independent Idea,
which we may designate by the term Polarity.
We shall trace some of the forms in which this Idea
has manifested itself in the history of science. In doing
so we shall not begin, as in other Books of this work
346 PHILOSOPHY OF THE MECHANICO-CIIEMICAL SCIENCES.
we have done, by speaking of the notion as it is em
ployed in common use : for the relation of polarity is of
so abstract and technical a nature, that it is not employed,
at least in any distinct and obvious manner, on any
ordinary or practical occasions. The idea belongs pecu
liarly to the region of speculation : in persons of com
mon habits of thought it is probably almost or quite
undeveloped ; and even most of those whose minds have
been long occupied by science, find a difficulty in appre
hending it in its full generality and abstraction, and
stript of all irrelevant hypothesis.
2. Magnetism, — The name and the notion of Poles
*J
were first adopted in the case of a magnet. If we have
two magnets, their extremities attract and repel each
other alternatively. If the first end of the one attract
the first end of the other, it repels the second end, and
conversely. In order to express this rule conveniently,
the two ends of each magnet are called the north pole
and the south pole respectively, the denominations being
borrowed from the poles of the earth and heavens.
"These poles," as Gilbert says'", ''regulate the motions
of the celestial spheres and of the earth. In like manner
the magnet has its poles, a northern and a southern one ;
certain and determined points constituted by nature in
the stone, the primary terms of its motions and effects,
the limits and governors of many actions and virtues."
The nature of the opposition of properties of which
we speak may be stated thus.
The North pole of one magnet attracts the South
pole of another magnet.
The North pole of one magnet repels the North pole
of another magnet.
The South pole of one magnet repels the South pole
of another magnet.
* DC Matin., Lib. i. c. iii.
APPLICATION OF THE IDEA OF POLARITY. 347
The South pole of one magnet attracts the North
pole of another magnet.
It will be observed that the contrariety of position
which is indicated by putting the South pole for the
North pole in either magnet, is accompanied by the
opposition of mechanical effect which is expressed by
changing attraction into repulsion and repulsion into
attraction : and thus we have the general feature of
polarity : — A contrast of properties corresponding to a
contrast of positions.
!J. Electricity. — When the phenomena of electricity
came to be studied, it appeared that they involved rela
tions in some respects analogous to those of magnetism.
Two kinds of electricity were distinguished, the
positive and the negative ; and it appeared that two
bodies electrized positively or two electrized negatively,
repelled each other, like two north or two south magnetic
poles ; while a positively and a negatively electrized body
attracted each other, like the north and south poles of
two magnets. In conductors of an oblong form, the
electricity could easily be made to distribute itself so
that one end should be positively and one end negatively
electrized ; and then such conductors acted on each other
exactly as magnets would do.
But in conductors, however electrized, there is no
peculiar point which can permanently be considered as
the pole. The distribution of electricity in the conduc
tor depends upon external circumstances : and thus,
although the phenomena offer the general character of
polarity — alternative results corresponding to alternative
positions,— they cannot be referred to poles. Some other
mode of representing the forces must be adopted than
that which makes them emanate from permanent points
as in a magnet.
The phenomena of attraction and repulsion in elec-
348 riiiLOSoniYoF THE MECIIANICO-CILEMICAL SCIENCES.
trized bodies were conveniently represented by means of
the hypothesis of tiro electric fluids, a positive and a
negative one, which were supposed to be distributed in
the bodies. Of these fluids, it was supposed that each
repelled its own parts and attracted those of the opposite
fluid : and it was found that this hypothesis explained all
the obvious laws of electric action. Here then we have
the phenomena of polarization explained by a new kind
of machinery : — two opposite fluids distributed in bodies,
and supplying them, so to speak, with their polar forces.
This hypothesis not only explains electrical attraction,
but also the electrical spark : when two bodies, of which
the neighbouring surfaces are charged with the two
opposite fluids, approach near to each other, the mutual
attraction of the fluids becomes more and more intense,
till at last the excess of fluid on the one body breaks
through the air and rushes to the other body, in a form
accompanied by light and noise. When this transfer has
taken place, the attraction ceases, the positive and the
negative fluid having neutralized each other. Their
effort was to unite ; and this union being effected, there
is no longer any force in action. Bodies in their natural
unexcited condition may be considered as occupied by a
combination of the two fluids : and hence we see how
the production of either kind of electricity is necessarily
accompanied with the production of an equivalent amount
of the opposite kind.
4. Voltaic Electricity. — Such is the case in Franklinic
electricity, — that which is excited by the common elec
trical machine. In studying Voltaic electricity, we are
led to the conviction that the fluid which is in a condi
tion of momentary equilibrium in electrized conductors,
exists in the state of current in the voltaic circuit. And
here we find polar relations of a new kind existing among
the forces. Two voltaic currents attract each other when
APPLICATION OF THE IDEA OF POLARITY. 34!)
they are moving in the same, and repel each other when
they are moving in opposite, directions.
But we find, in addition to these, other polar rela
tions of a more abstruse kind, and which the supposition
of two fluids does not so readily explain. For instance,
if such fluids existed, distinct from each other, it might
be expected that it would be possible to exhibit one
of them separate from the other. Vet in all the phe
nomena of electromotive currents, we attempt in vain
to obtain one kind of electricity separately. "I have
not," says Mr. Faraday'", "been able to find a single
fact which could be adduced to prove the theory of
two electricities rather than one, in electric currents;
or, admitting the hypothesis of two electricities, have
I been able to perceive the slightest grounds that one
electricity can be more powerful than the other, — or
that it can be present without the other, — or that it
can be varied or in the slightest degree affected without
a corresponding variation in the other." " Thus," he
adds, " the polar character of the powers is rigorous and
complete." Thus, we too may remark, all the super
fluous and precarious parts gradually drop off from the
hypothesis which we devise in order to represent polar
phenomena; and the abstract notion of polarity — of equal
and opposite powers called into existence by a com
mon condition — remains unincumbered with extraneous
machinery.
5. Light. — Another very important example of the
application of the idea of polarity is that supplied by the
discovery of the polarization of light. A ray of light
may, by various processes, be modified, so that it has dif
ferent properties according to its different sides, although
this difference is not perceptible by any common effects.
If, for instance, a ray thus modified, pass perpendicularly
* Rexearches, ;">](>.
350 PHILOSOPHY OF THE MECIIANICO-CHEMICAL SCIENCES.
through a circular glass, and fall upon the eye, we may
turn the glass round and round its frame, and we shall
made no difference in the brightness of the spot which
we see. But if, instead of a glass, we look through a
longitudinal slice of tourmaline, the spot is alternately
dark and bright as we turn the crystal through successive
quadrants. Here we have a contrast of properties (dark
and bright) corresponding to a contrast of positions, (the
position of a line east and west being contrasted with
the position north and south,) which, as we have said, is
the general character of polarity. It was with a view of
expressing this character that the term polarization was
originally introduced. Mains was forced by his disco
veries into the use of this expression. " We find," he
says, in 1811, "that light acquires properties which are
relative only to the sides of the ray, — which are the same
for the north and south sides of the ray, (using the
points of the compass for description's sake only,) and
which are different when we go from the north and south
to the east or to the west sides of the ray. I shall give
the name of poles to these sides of the ray, and shall call
polarization the modification which gives to light these
properties relative to these poles. I have put off hitherto
the admission of this term into the description of the
physical phenomena with which we have to do : I did
not dare to introduce it into the Memoirs in which I
published my last observations : but the variety of forms
in which this new phenomenon appears, and the difficulty
of describing them, compel me to admit this new expres
sion ; which signifies simply the modification which light
has undergone in acquiring new properties which are not
relative to the direction of the ray, but only to its sides
considered at right angles to each other, and in a plane
perpendicular to its direction."
The theory which represents light as an emission of
APPLICATION OF THE IDEA OF POLARITY. 35 1
particles was in vogue at the time when Mains published
his discoveries; and some of his followers in optical
research conceived that the phenomena which he thus
described rendered it necessary to ascribe poles and an
axis to each particle of light. On this hypothesis, light
would be polarized when the axes of all the particles
were in the same direction : and, making such a suppo
sition, it may easily be conceived capable of transmission
through a crystal whose axis is parallel to that of the
luminous particles, and intransmissible when the axis of
the crystal is in a position transverse to that of the par
ticles.
The hypothesis of particles possessing poles is a rude
and arbitrary assumption, in this as in other cases ; but
it serves to convey the general notion of polarity, which
is the essential feature of the phenomena. The term
"polarization of light" has sometimes been complained
of in modern times as hypothetical and obscure. But the
real cause of obscurity was, that the Idea of Polarity was,
till lately, very imperfectly developed in men's minds.
As we have seen, the general notion of polarity, — oppo
site properties in opposite directions, — exactly describes
the character of the optical phenomena to which the
term is applied.
It is to be recollected that in optics we never speak
of the poles, but of the plane of polarization of a ray. The
word sides, which Newton and Mains have used, neither
of them appears to have been satisfied with ; Newton, in
employing it, had recourse to the strange Gallicism of
speaking of the coast of usual and of unusual refraction
of a crystal.
The modern theory of optics represents the plane of
polarization of light as depending, not on the position in
which the axes of the luminifcrous particles lie, but on
the direction of those transverse vibrations in which light
352 PHILOSOPHY OF THE MECHANICO-CHEMICAL SCIENCES.
consists. This theory is, as we have stated in the His
tory, recommended by an extraordinary series of suc
cesses in accounting for the phenomena. And this
hypothesis of transverse vibrations shows us another
mechanical mode, (besides the hypothesis of particles
with axes,) by which we may represent the polarity of a
ray. But we may remark that the general notion of
polarity, as applied to light in such cases, would subsist,
even if the undulatory theory were rejected. The idea
is, as we have before said, independent of all hypothetical
machinery.
I need not here refer to the various ways in which
light may be polarized, as, for instance, by being reflected
from the surface of water or of glass at certain angles, by
being transmitted through crystals, and in other ways.
In all cases the modification produced, the polarization,
is identically the same property. Nor need I mention
the various kinds of phenomena which appear as contrasts
in the result ; for these are not merely light and dark, or
white and black, but red and green, and generally, a
colour and its complementary colour, exhibited in many
complex and varied configurations. These multiplied
modes in which polarized light presents itself add nothing
to the original conception of polarization : and I shall
therefore pass on to another subject.
G. Crystallization. — Bodies which are perfectly crys
tallized exhibit the most complete regularity and sym
metry of form ; and this regularity not only appears in
their outward shape, but pervades their whole texture,
and manifests itself in their cleavage, their transparency,
and in the uniform and determinate optical properties
which exist in every part, even the smallest fragment of
the mass. If we conceive crystals as composed of par
ticles, we must suppose these particles to be arranged in
the most regular manner ; for example, if we suppose
APPLICATION OF THE IDEA OF POLARITY. 353
each particle to have an axis, we must suppose all these
axes to be parallel ; for the direction of the axis of the
particles is indicated by the physical and optical pro
perties of the crystal, and therefore this direction must
be the same for every portion of the crystal. This
parallelism of the axes of the particles may be con
ceived to result from the circumstance of each particle
having poles, the opposite poles attracting each other.
In virtue of forces acting as this hypothesis assumes, a
collection of small magnetic particles would arrange
themselves in parallel positions ; and such a collection of
magnetic particles offers a sort of image of a crystal.
Thus we are led to conceive the particles of crystals as
polarized, and as determined in their crystalline positions
by polar forces. This mode of apprehending the consti
tution of crystals has been adopted by some of our most
eminent philosophers. Thus Berzelius says""', "It is de
monstrated, that the regular forms of bodies presuppose
an effort of their atoms to touch each other by preference
in certain points ; that is, they are founded upon a Pola
rity ;" — he adds, " a polarity which can be no other than
an electric or magnetic polarity." In this latter clause
we have the identity of different kinds of polarity
asserted ; a principle which we shall speak of in the
next chapter. But we may remark, that even without
dwelling upon this connexion, any notion which we can
form of the structure of crystals necessarily involves the
idea of polarity. Whether this polarity necessarily re
quires us to believe crystals to be composed of atoms
which exert an effort to touch each other in certain points
by preference, is another question. And, in agreement
with what has been said respecting other kinds of polarity,
we shall probably find, on a more profound examination
of the subject, that while the idea of polarity is essential,
::" Essay on the T/ieori/ of Chemical Properties, 15520, j>. \\\\.
VOL. I. W. P. A A
354 PHILOSOPHY OF THE MECHANICO-CHEMICAL SCIENCES.
the machinery by which it is thus expressed is precarious
and superfluous.
7. Chemical Affinity. — We shall have, in the next
Book, to speak of Chemical Affinity at some length ; but
since the ultimate views to which philosophers have been
led, induce them to consider the forces of affinity as
polar forces, we must enumerate these among the exam
ples of polarity. In chemical processes, opposites tend
to unite, and to neutralize each other by their union.
Thus an acid or an alkali combine with vehemence, and
form a compound, a neutral salt, which is neither acid
nor alkaline.
This conception of contrariety and mutual neutraliza
tion, involves the idea of polarity. In the conception, as
entertained by the earlier chemists, the idea enters very
obscurely : but in the attempts which have more recently
been made to connect this relation (of acid and base,) with
other relations, the chemical elements have been conceived
as composed of particles which possess poles ; like poles
repelling, and unlike attracting each other, as they do in
magnetic and electric phenomena. This is, however, a
rude and arbitrary way of expressing polarity, and, as may
be easily shown, involves many difficulties which do not
belong to the idea itself. Mr. Faraday, who has been
led by his researches to a conviction of the polar nature
of the forces of chemical affinity, has expressed their
character in a more general manner, and without any of
the machinery of particles indued with poles. Accord
ing to his view, chemical synthesis and analysis must
always be conceived as taking place in virtue of equal
and opposite forces, by which the particles are united or
separated. These forces, by the very circumstance of
their being polar, may be transferred from point to point.
For if we conceive a string of particles, and if the positive
force of the first particle be liberated and brought into
APPLICATION OF THE IDEA OF POLARITY. :}.">.">
action, its negative force also must be set free : this
negative force neutralizes the positive force of the next
particle, and therefore the negative force of this particle
(before employed in neutralizing its positive force,) is set
free : this is in the same way transferred to the next
particle, and so on. And thus we have a positive force
active at one extremity of a line of particles, correspond
ing to a negative force at the other extremity, all the
intermediate particles reciprocally neutralizing each
other's action. This conception of the transfer of chemi
cal action was indeed at an earlier period introduced by
Grotthus"-", and confirmed by Davy. But in Mr. Fara
day's hands we see it divested of all that is superfluous,
and spoken of, not as a line of particles, but as *' an axis
of power, having [at every point,] contrary forces, ex
actly equal, in opposite directions."
8. General Remarks. — Thus, as we see, the notion
of polarity is applicable to many large classes of phe
nomena. Yet the idea in a distinct and general form is
only of late growth among philosophers. It has gra
dually been abstracted and refined from many extraneous
hypotheses which were at first supposed to be essential
to it. We have noticed some of these hypotheses ; — as
the poles of a body; the poles of the particles of a fluid ;
two opposite fluids; a single fluid in excess and defect;
transverse vibrations. To these others might be added.
Thus Dr. Proutf assumes that the polarity of molecules
results from their rotation on their axes, the opposite
motions of contiguous molecules being the cause of
opposite (positive and negative) polarities.
But none of these hypotheses can be proved by the
fact of polarity alone ; and they have been in succession
rejected when they had been assumed on that ground.
* DUMAS, Lcqons sur la Philosophy Chimiqite, p. 401.
t Bridgcwater Treatixe, p. .r>.~>i).
A A "2
3~>(> PHILOSOPHY OF THE MECHANICO-CHEMICAL SCIENCES.
Thus Davy, in 18*20, speaking of chemical forces says*,
•' In assuming the idea of two ethereal, subtile, elastic
fluids, attractive of the particles of each other, and
repulsive as to their own particles, capable of combining
in different proportions with bodies, and according to
their proportions giving them their specific qualities and
rendering them equivalent masses, it would be natural
to refer the action of the poles to the repulsions of the
substances combined with the excess of one fluid, and
the attractions of those united to the excess of the other
fluid; and a history of the phenomena, not unsatisfactory
to the reason, might in this way be made out. But as
it is possible likewise to take an entirely different view
of the subject, on the idea of the dependence of the
results upon the primary attractive powers of the parts
of the combination on a single subtile fluid, I shall not
enter into any discussion on this obscure part of the
theory." Which of these theories will best represent the
case, will depend upon the consideration of other facts,
in combination with the polar phenomena, as we see in
the history of optical theory. In like manner Mr.
Faraday proved by experiment f the errour of all theories
which ascribe electro-chemical decomposition to the
attraction of the poles of the voltaic battery.
In order that they may distinctly image to them
selves the idea of polarity, men clothe it in some of
the forms of machinery above spoken of; yet every new
attempt shows them the unnecessary difficulties in which
they thus involve themselves. But on the other hand
it is difficult to apprehend this idea divested of all
machinery ; and to entertain it in such a form that it
shall apply at the same time to magnetism and elec
tricity, galvanism and chemistry, crystalline structure
and light. The Idea of Polarity becomes most pure and
- Phil. 7V., 182(5, p. 41.1. I Itwarc/ies, p. 495, &c.
APPLICATION OF THE IDEA OF POLARITY. of) 7
genuine, when we entirely reject the conception of Poles,
as Faraday has taught us to do in considering electro
chemical decomposition ; but it is only by degrees and
by effort that we can reach this point of abstraction and
generality.
9. There is one other remark which we may here
make. It was a maxim commonly received in the ancient
schools of philosophy, that " like attracts like :" but as
we have seen, the universal maxim of polar phenomena
is, that like repels like, and attracts unlike. The north
pole attracts the south pole, the positive fluid attracts
the negative fluid ; opposite elements rush together ;
opposite motions reduce each other to rest. The per
manent and stable course of things is that which results
from the balance and neutralization of contrary ten
dencies. Nature is constantly labouring after repose by
the effect of such tendencies ; and so far as polar forces
enter into her economy, she seeks harmony by means of
discord, and unity by opposition.
Although the Idea of Polarity is as yet somewhat
vague and obscure, even in the minds of the cultivators
of physical science, it has nevertheless given birth to
some general principles which have been accepted as
evident, and have had great influence on the progress
of science. These we shall now consider.
CHAPTER II.
OF THE CONNEXION OF POLARITIES.
1. IT has appeared in the preceding chapter that in
cases in which the phenomena suggest to us the idea of
polarity, we are also led to assume some material ma
chinery as the mode in which the polar forces are exerted.
358 PHILOSOPHY OF Till-: MECHANICO-CHEMICAL SCIENCES.
We assume, for instance, globular particles which possess
poles, or the vibrations of a fluid, or two fluids attract
ing each other ; in every case, in short, some hypothesis
by which the existence and operation of the polarity is
embodied in geometrical and mechanical properties of a
medium ; nor is it possible for us to avoid proceeding
upon the conviction that some such hypothesis must be
true ; although the nature of the connexion between
the mechanism and the phenomena must still be inde
finite and arbitrary.
But since each class of polar phenomena is thus
referred to an ulterior cause, of which we know no more
than that it has a polar character, it follows that different
polarities may result from the same cause manifesting
its polar character under different aspects. Taking, for
example, the hypothesis of globular particles, if elec
tricity result from an action dependent upon the poles
of each globule, magnetism may depend upon an action
in the equator of each globule; or taking the supposition
of transverse vibrations, if polarized light result directly
from such vibrations, crystallization may have reference
to the axes of the elasticity of the medium by which the
vibrations are rendered transverse, — so far as the polar
character only of the phenomena is to be accounted for.
I say this may be so, in so far only as the polar cha
racter of the phenomena is concerned ; for whether the
relation of electricity to magnetism, or of crystalline
forces to light, can really be explained by such hypo
theses, remains to be determined by the facts themselves.
But since the first necessary feature of the hypothesis
is, that it shall give polarity, and since an hypothesis
which does this, may, by its mathematical relations, give
polarities of different kinds and in different directions,
any two co-existent kinds of polarity may result from
the same cause, manifesting itself in various manners.
OF THE CONNEXION OF POLARITIES. 350
The conclusion to which we are led by these general
considerations is, that two co-existing classes of polar
phenomena may be effects of the same cause. But those
who have studied such phenomena more deeply and
attentively have, in most or in all cases, arrived at the
conviction that the various kinds of polarity in such
cases must be connected and fundamentally identical.
As this conviction has exercised a great influence, both
upon the discoveries of new facts and upon the theore
tical speculations of modern philosophers, and has been
put forward by some writers as a universal principle of
science, I will consider some of the cases in which it has
been thus applied.
'2. Connexion of Magnetic and Electric Polarity. —
The polar phenomena of electricity and magnetism are
clearly analogous in their laws: and obvious facts showed
at an early period that there was some connexion be
tween the two agencies. Attempts were made to esta
blish an evident and definite relation between the two
kinds of force, which attempts proceeded upon the prin
ciple now under consideration ; — namely, that in such
cases, the two kinds of polarity must be connected. Pro
fessor (Ersted, of Copenhagen, was one of those who
made many trials founded upon this conviction : yet all
these were long unsuccessful. At length, in 1820, he
discovered that a galvanic current, passing at right angles
near to a magnetic needle, exercises upon it a powerful
deflecting force. The connexion once detected between
magnetism and galvanism was soon recognized as con
stant and universal. It was represented in different
hypothetical modes by different persons ; some consider
ing the galvanic current as the primitive axis, and the
magnet as constituted of galvanic currents passing round
it at right angles to the magnetic axis; while others
conceived the magnetic axis as the primitive one, and
o()0 PHILOSOPHY OF Till: MECHANICO-CHEMICAL SCIENCES.
the electric current as implying a magnetic current
round the wire. So far as many of the general relations
of these two kinds of force were concerned, either mode
of representation served to express them ; and thus the
assumption that the two polarities, the magnetic and
the electric, were fundamentally identical, was verified,
so far as the phenomena of magnetic attraction, and the
like, were concerned.
I need not here mention how this was further con
firmed by the experiments in which, by means of the
forces thus brought into view, a galvanic wire was made
to revolve round a magnet, and a magnet round a gal
vanic wire ; — in which artificial magnets were constructed
of coils of galvanic wire ; — and finally, in which the gal
vanic spark was obtained from the magnet. The identity
which sagacious speculators had divined even before it
was discovered, and which they had seen to be universal
as soon as it was brought to light, was completely mani
fested in every imaginable form.
The relation of the electric and magnetic polarities
was found to be, that they were transverse to each
other, and this relation exhibited under various condi
tions of form and position of the apparatus, gave rise to
very curious and unexpected perplexities. The degree
of complication which this relation may occasion, may be
judged of from the number of constructions and modes
of conception offered by CErsted, Wollaston, Faraday,
and others, for the purpose of framing a technical memory
of the results. The magnetic polarity gives us the north
and south poles of the needle ; the electric polarity
makes the current positive and negative; and these pairs
of opposites are connected by relations of situation, as
above and below, right and left ; and give rise to the
resulting motion of the needle one way or the other.
3. Ampere, by framing his hypotheses of the action
OF THE CONNEXION OF POLARITIES. 301
of voltaic currents and the constitution of magnets,
reduced all these technical rules to rigorous deductions
from one general principle. And thus the vague and
obscure persuasion that there must be some connexion
between electricity and magnetism, so long an idle and
barren conjecture, was unfolded into a complete theory,
according to which magnetic and electromotive actions
are only two different manifestations of the same forces;
and all the above-mentioned complex relations of pola
rities are reduced to one single polarity, that of the
electro-dynamic current.
4. As the idea of polarity was thus firmly established
and clearly developed, it became an instrument of rea
soning. Thus it led Ampere to maintain that the original
or elementary forces in electro-dynamic action could not
be as M. Biot thought they were, a statical couple, but
must be directly opposite to each other. The same idea
enabled Mr. Faraday to carry on with confidence such
reasonings as the following'"" : " No other known power
has like direction with that exerted between an electric
current and a magnetic pole ; it is tangential, while all
other forces acting at a distance are direct. Hence if a
magnetic pole on one side of a revolving plate follow
its course by reason of its obedience to the tangential
force exerted upon it by the very current of electricity
which it has itself caused ; a similar pole on the other
side of the plate should immediately set it free from this
force ; for the currents which have to be formed by the
two poles are in contrary directions." And in Article
1114 of his Researches, the same eminent philosopher
infers that if electricity and magnetism are considered
as the results of a peculiar agent or condition, exerted
in determinate directions perpendicular to each other,
one must be by some means convertible into the other;
" Researches. 244.
:}()'2 PHILOSOPHY OF THE MECHANICO-CHEM1CAL SCIENCES.
and this he was afterwards able to prove to be the case
in fact.
Thus the principle that the co-existent polarities of
magnetism and electricity are connected and fundamen
tally identical, is not only true, but is far from being-
cither vague or barren. It has been a fertile source
both of theories which have, at present, a very great pro
bability, and of the discovery of new and striking facts.
We proceed to consider other similar cases.
5. Connexion of Electrical and Chemical Polari
ties. — The doctrine that the chemical forces by which
the elements of bodies are held together or separated,
are identical with the polar forces of electricity, is a
great discovery of modern times ; so great and so recent,
indeed, that probably men of science in general have
hardly yet obtained a clear view and firm hold of this
truth. This doctrine is now, however, entirely esta
blished in the minds of the most profound and philoso
phical chemists of our time. The complete developemerit
and confirmation of this as of other great truths, was
preceded by more vague and confused opinions gradu
ally tending to this point; and the progress of thought
and of research was impelled and guided, in this as in
similar cases, by the persuasion that these co-existent
polarities could not fail to be closely connected with
each other. While the ultimate and exact theory to
which previous incomplete and transitory theories tended
is still so new and so unfamiliar, it must needs be a
matter of difficulty and responsibility for a common
reader to describe the steps by which truth has advanced
from point to point. I shall, therefore, in doing this,
guide myself mainly by the historical sketches of the
progress of this great theory, which, fortunately for us,
have been given us by the two philosophers who have
OF THE CONNEXION OF 1'OLAUITIES. JJG3
played by far the most important parts in the discovery,
Davy and Faraday.
It will be observed that we are concerned here with
the progress of theory, and not of experiment, except so
far as it is confirmatory of theory. In Davy's Memoir*
of 1826, on the Relations of Electrical and Chemical
Changes, he gives the historical details to which I have
alluded. Already in 1802 he had conjectured that all
chemical decompositions might be polar. In 1806 he
attempted to confirm this conjecture, and succeeded, to
his own satisfaction, in establishing + that the combina
tions and decompositions by electricity were referable
to the law of electrical attractions and repulsions ; and
advanced the hypothesis (as he calls it,) that chemical
and electrical attractions were produced by the same
cause, acting in one case on particles, in the other on
masses. This hypothesis was most strikingly confirmed
by the author's being able to use electrical agency as a
more powerful means of chemical decomposition than
any which had yet been applied. " Believing," he adds,
"that our philosophical systems are exceedingly im
perfect, I never attached much importance to this hypo
thesis; but having formed it after a copious induction
of facts, and having gained by the application of it a
number of practical results, and considering myself as
much the author of it as I was of the decomposition of
the alkalies, and having developed it in an elementary
work as far as the present state of chemistry seemed to
allow, I have never," he says "criticized or examined
the manner in which different authors have adopted or
explained it, contented, if in the hands of others, it
assisted the arrangements of chemistry or mineralogy,
or became an instrument of discovery." When the doc
trine had found an extensive acceptance among chemists,
* Phil. Trans., 1826, p. 383. * Ibid., p. 380.
r>(>4 PHILOSOPHY OF THE MECHANICO-CHEMICAL SCIENCES.
attempts were made to show that it had been asserted
by earlier writers : and though Davy justly denies all
value to these pretended anticipations, they serve to
show, however dimly, the working of that conviction of
the connexion of co-existent properties which all along
presided in men's minds during this course of investi
gation. " Hitter and Winterl have been quoted," Davy
says*, "•among other persons, as having imagined or
anticipated the relation between electrical powers and
chemical affinities before the discovery of the pile of
Volta. But whoever will read with attention Ritter's
' Evidence that Galvanic action exists in organized
nature,' and Winter's Prolusiones ad Chemiam sceculi
decimi noni, will find nothing to justify this opinion."
He then refers to the Queries of Xewton at the end of
his Optics. " These," he says, " contain more grand and
speculative views that might be brought to bear upon
this question than any found in the works of modern
electricians ; but it is very unjust to the experimentalists
who by the laborious application of new instruments,
have discovered novel facts and analogies, to refer them
to any such suppositions as that all attractions, chemical,
electrical, magnetical, and gravitative, may depend upon
the same cause." It is perfectly true, that such vague
opinions, though arising from that tendency to generalize
which is the essence of science, are of no value except
so far as they are both rendered intelligible, and con
firmed by experimental research.
The phenomena of chemical decomposition by means
of the voltaic pile, however, led other persons to views
very similar to those of Davy. Thus Grotthus in 1805 f
published an hypothesis of the same kind. " The pile of
Volta," he says, " is an electrical magnet, of which each
element, that is, each pair of plates, has a positive and a
* Phil. Trans., 1826. p. 384. t Ann. C/iim.. Lxviii. 54.
OK THE CONNEXION OF POLARITIES. 305
negative pole. The consideration of this polarity sug
gested to me the idea that a similar polarity may come
into play between the elementary particles of water
when acted upon by the same electrical agent ; and I
avow that this thought was for me a flash of light."
0. The thought, however, though thus brought into
being, was very far from being as yet freed from vague
ness, superfluities, and errours. I have elsewhere noticed*
Faraday's remark on Davy's celebrated Memoir of 1806;
that " the mode of action by which the effects take place
is stated very generally, so generally, indeed, that pro
bably a dozen precise schemes of electro-chemical action
might be drawn up, differing essentially from each other,
yet all agreeing with the statement there given." When
Davy and others proceeded to give a little more defi-
niteness and precision to the statement of their views,
they soon introduced into the theory features which it
was - afterwards found necessary to abandon. Thusf
both Davy, Grotthus, Riffault, and Chompre, ascribed
electrical decomposition to the action of the poles, and
some of them even pretended to assign the proportion
in which the force of the pole diminishes as the distance
from it increases. Faraday, as I have already stated,
showed that the polarity must be considered as residing
not only in what had till then been called the poles,
but at every point of the circuit. He ascribed j electro
chemical decomposition to internal forces, residing in
the particles of the matter under decomposition, not to
external forces, exerted by the poles. Hence he shortly
afterwards § proposed to reject the word poles altogether,
and to employ instead, the term electrode, meaning the
* Hist. Ind. Sci., B. xiv. c. ix. sect. 1.
t See Faraday's Historical Sketch, Researches, 4H1 — 402.
+ Art 524.
§ In 1834. l.lcventh Series of Researches. Art. (502.
:if)(> PHILOSOPHY OF THE MECHANICO-CHEMICAL SCIKXCKS.
doors or passages (of whatever surface formed,) by which
the decomposed elements pass out. What have been
called the positive and negative poles he further termed
the anode and cathode \ and he introduced some other
changes in nomenclature connected with these. He
then, as I have related in the History"", invented the
Volta-electrometer, which enabled him to measure the
quantity of voltaic action, and this he found to be iden
tical with the quantity of chemical affinity; and he was
thus led to the clearest view of the truth towards which
he and his predecessors had so long been travelling,
that electrical and chemical forces are identical!.
7. It will, perhaps, be said that this beautiful train
of discovery was entirely due to experiment, and not to
any a priori conviction that co-existent polarities must
be connected. I trust I have sufficiently stated that
such an a priori principle could not be proved, nor even
understood, without a most laborious and enlightened
use of experiment ; but yet I think that the doctrine
when once fully unfolded, exhibited clearly, and estab
lished as true, takes possession of the mind with a more
entire conviction of its certainty and universality, in
t/ V *
virtue of the principle we are now considering. When
the theory has assumed so simple a form, it appears to
derive immense probability (to say the least) from its
simplicity. Like the laws of motion, when stated in its
most general form, it appears to carry with it its own
evidence. And thus this great theory borrows some
thing of its character from the Ideas which it involves,
as well as from the Experiments by which it was esta
blished.
8. We may find in many of Mr. Faraday's subsequent
reasonings, clear evidence that this idea of the connex
ion of polarities, as now developed, is not limited in its
* Hist. Iml. Set., B. xiv. o. ix. sect 2. + Arts. 915, 916, 91?.
OF THE CONNEXION OF POLARITIES. 307
application to facts already known experimentally, hut,
like other ideas, determines the philosopher's researches
into the unknown, and gives us the form of knowledge
even before we possess the matter. Thus, he says, in
his Thirteenth Series*, "I have long sought, and still
seek, for an effect or condition which shall be to statical
electricity what magnetic force is to current electricity ;
for as the lines of discharge are associated with a cer
tain transverse effect, so it appeared to me impossible
but that the lines of tension or of inductive action,
which of necessity precede the discharge, should also
have their correspondent transverse condition or effect."
Other similar passages might be found.
I will now consider another case to which we may
apply the principle of connected polarities.
9. Connexion of Chemical and Crystalline Polari
ties. — The close connexion between the chemical affinity
and the crystalline attraction of elements cannot be
overlooked. Bodies never crystallize but when their
elements combine chemically ; and solid bodies which
combine, when they do it most completely and exactly,
also crystallize. The forces which hold together the ele
ments of a crystal of alum are the same forces which
make it a crystal. There is no distinguishing between
the two sets of forces.
Both chemical and crystalline forces are polar, as we
stated in the last chapter; but the polarity in the two
cases is of a different kind. The polarity of chemical
forces is then put in the most distinct form, when it is
identified with electrical polarity ; the polarity of the
particles of crystals has reference to their geometrical
form. And it is clear that these two kinds of polarity
must be connected. Accordingly, Berzelius expressly
asserts t the necessary identity of these two polarities.
* Art. 16f>8. t Essay on Chemical Prop., 113.
o()8 PHILOSOPHY OF THE MECHANICO-CHEMICAL SCIENCES.
" The regular forms of 'bodies suppose a polarity which
can be no other than an electric or magnetic polarity."
This being so seemingly inevitable, we might expect to
find the electric forces manifesting some relation to the
definite directions of crystalline forms. Mr. Faraday
tried, but in vain, to detect some such relation. He
attempted to ascertain* whether a cube of rock crystal
transmitted the electrical force of tension with different
intensity along and across the axis of the crystal. In
the first specimen there seemed to be some difference ;
but in other experiments, made both with rock crystal
and with calc spar, this difference disappeared. Al
though therefore we may venture to assert that there
O *J
must be some very close connexion between electrical
and crystalline forces, we are, as yet, quite ignorant
what the nature of the connexion is, and in what kind
of phenomena it will manifest itself.
10. Connexion of Crystalline and Optical Polarities.
—Crystals present to us optical phenomena which have
a manifestly polar character. The double refraction,
both of uniaxal and of biaxal crystals, is always accom
panied with opposite polarization of the two rays ; and
in this and in other ways light is polarized in directions
dependent upon the axes of the crystalline form, that is,
on the directions of the polarities of the crystalline par
ticles. The identity of these two kinds of polarity (cry
stalline and optical) is too obvious to need insisting on ;
and it is not necessary for us here to decide by what
hypothesis this identity may most properly be repre
sented. We may hereafter perhaps find ourselves jus
tified in considering the crystalline forces as determining
the elasticity of the luminiferous ether to be different
in different directions within the crystal, and thus as
determining the refraction and polarization of the light
* Rt'sean-Jics. Art. 1689.
OF THE CONNEXION OF POLARITIES. 3()0
which the crystal transmits. But at present we merely
note this case as an additional example of the manifest
connexion and fundamental identity of two co-existent
polarities.
11. Connexion of Polarities in general. — Thus we
find that the connexion of different kinds of polarities,
magnetic, electric, chemical, crystalline, and optical, is
certain as a truth of experimental science. We have
attempted to show further that in the minds of several
of the most eminent discoverers and philosophers, such
a conviction is something more than a mere empirical
result: it is a principle which has regulated their re
searches while it was still but obscurely seen and imper
fectly unfolded, and has given to their theories a charac
ter of generality and self-evidence which experience
alone cannot bestow.
It will, perhaps, be said that these doctrines, — that
scientific researches may usefully be directed by prin
ciples in themselves vague and obscure ; — that theories
may have an evidence superior to and anterior to expe
rience ; — are doctrines in the highest degree dangerous,
and utterly at variance with the soundest maxims of
modern times respecting the cultivation of science.
To the justice and wisdom of this caution I entirely
agree : and although I have shown that this principle of
the connexion of polarities, rightly interpreted and esta
blished in each case by experiment, involves profound
and comprehensive truths ; I think it no less important
to remark that, at least in the present stage of our
knowledge, we can make no use of this principle with
out taking care, at every step, to determine by clear and
decisive experiments, its proper meaning and applica
tion. All endeavours to proceed otherwise have led,
and must lead, to ignorance and confusion. Attempts
to deduce from our bare idea of polarity, and our fun-
VOL i. w. r. B B
370 PHILOSOPHY OF THE MECHAN1CO-CHEMICAL SCIENCES.
damental convictions respecting the connexion of polari
ties, theories concerning the forces which really exist in
nature, can hardly have any other result than to bewilder
men's minds, and to misdirect their efforts.
So far, indeed, as this persuasion of a connexion
among apparently different kinds of agencies, impels
men, engaged in the pursuit of knowledge, to collect
observations, to multiply, repeat, and vary experiments,
and to contemplate the result of these in all aspects
and relations, it may be an occasion of the most impor
tant discoveries. Accordingly we find that the great
laws of phenomena which govern the motions of the
planets about the sun, were first discovered by Kepler,
in consequence of his scrutinizing the recorded observa
tions with an intense conviction of the existence of geo
metrical and arithmetical harmonies in the solar system.
Perhaps we may consider the discovery of the connexion
of magnetism and electricity by Professor QErsted in
1820, as an example somewhat of the same kind; for
he also was a believer in certain comprehensive but un
defined relations among the properties of bodies ; and
in consequence of such views entertained great admira
tion for the Prologue to the Chemistry of the Nineteenth
Century, of Winterl, already mentioned. M. (Ersted, in
1803, published a summary of this work ; and in so do
ing, praised the views of Winterl as far more profound
and comprehensive than those of Lavoisier. Soon after
wards a Review of this publication appeared in France **,
in which it was spoken of as a work only fit for the
dark ages, and as the indication of a sect which had
for some time " ravaged Germany," and inundated that
country with extravagant and unintelligible mysticism.
It was, therefore, a kind of triumph to M. (Ersted to
bo, after some years' labour, the author of one of the
* Ann. Chim., Tom. t. (1804), p. 191.
OF THE CONNEXION OK POLARITIES. :571
most remarkable and fertile physical discoveries of his
time.
12. It was not indeed without some reason that cer
tain of the German philosophers were accused of dealing
in doctrines vast and profound in their aspect, but, in
reality, indefinite, ambiguous, and inapplicable. And
the most prominent of such doctrines had reference to
the principle now under our consideration ; they repre
sented the properties of bodies as consisting in certain
polarities, and professed to deduce, from the very nature
of things, with little or no reference to experiment, the
existence and connexion of these polarities. Thus Schel-
ling, in his Ideas towards a Philosophy of Nature, pub
lished in 1803, says*, "Magnetism is the universal act
of investing Multiplicity with Unity ; but the universal
form of the reduction of Multiplicity to Unity is the
Line, pure Longitudinal Extension : hence Magnetism
is determination of pure Longitudinal Extension ; and
as this manifests itself by absolute Cohesion, Magnetism
is the determination of absolute Cohesion." And as
Magnetism was, by such reasoning, conceived to be
proved as a universal property of matter, Schelling as
serted it to be a confirmation of his views when it was
discovered that other bodies besides iron are magnetic.
In like manner he used such expressions as the follow-
ingf: "The threefold character of the Universal, the
Particular, and the Indifference of the two, — as ex
pressed in their Identity, is Magnetism, as expressed
in their Difference, is Electricity, and as expressed in
the Totality, is Chemical Process. Thus these forms
are only one form ; and the Chemical Process is a mere
transfer of the three Points of Magnetism into the Tri
angle of Chemistry."
It was very natural that the chemists should refuse
* P. 2'2'A. + P. -J8<).
It IJ '-'
372 PHILOSOPHY or THE MECIIANICO-CIIEMICAL SCIENCES.
to acknowledge, in this fanciful and vague language,
(delivered, however, it is to be recollected, in 1803,) an
anticipation of Davy's doctrine of the identity of electri
cal and chemical forces, or of (Ersted's electro-magnetic
agency. Yet it was perhaps no less natural that the
author of such assertions should look upon every great
step in the electro-chemical theory as an illustration
of his own doctrines. Accordingly we find Schelling
welcoming, with a due sense of their importance, the
discoveries of Faraday. When he heard of the experi
ment in which electricity was produced from common
magnetism, he fastened with enthusiasm upon the dis
covery, even before he knew any of its details, and pro
claimed it at a public meeting of a scientific body* as
one of the most important advances of modern science.
We have (he thus reasoned) three effects of polar forces ;
— electro-chemical Decomposition, electrical Action,
Magnetism. Volta and Davy had confirmed experimen
tally the identity of the two former agencies : (Ersted
showed that a closed voltaic circuit acquired magnetic
properties : but in order to exhibit the identity of elec
tric and magnetic action it was requisite that electric
forces should be extricated from magnetic. This great
step Faraday, he remarked, had made, in producing the
electric spark by means of magnets.
13. Although conjectures and assertions of the kind
thus put forth by Schelling involve a persuasion of the
pervading influence and connexion of polarities, which
persuasion has already been confirmed in many instances,
they involve this principle in a manner so vague and
ambiguous that it can rarely, in such a form, be of
any use or value. Such views of polarity can never
teach us in what cases we are and in what we are not
to expect to find polar relations ; and indeed tend rather
* Ucbor Faradav's Ncnesle Enldeckiins. Miinchen. 1832.
OF THE CONNEXION OF POLARITIES. 373
to diffuse error and confusion, than to promote know
ledge. Accordingly we cannot be surprized to find such
doctrines put forward by their authors as an evidence of
the small value and small necessity of experimental
science. This is done by the celebrated metaphysician
Hegel, in his Encyclopaedia*. '"Since," says he, "the
plane of incidence and of reflection in simple reflection
is the same plane, when a second reflector is introduced
which further distributes the illumination reflected from
the first, the position of the first plane with respect to
the second plane, containing the direction of the first
reflection and of the second, has its influence upon the
position, illumination or darkening of the object as it
appears by the second reflection. This influence must
be the strongest when the two planes are what we must
call negatively related to each other: — that is, when
they are at right angles." " But," he adds, " when men
infer (as Mai us has done) from the modification which
is produced by this situation, in the illumination of the
reflection, that the molecules of light in themselves,
that is, on their different sides, possess different physical
energies ; and when on this foundation, along with the
phenomena of entoptical colours therewith connected, a
wide labyrinth of the most complex theory is erected ;
we have then one of the most remarkable examples of
the inferences of physics from experiment." If Hegel's
reasoning prove anything, it must prove that polariza
tion always accompanies reflection under such circum
stances as he describes : yet all physical philosophers
know that in the case of metals, in which the reflection
is most complete, light is not completely polarized at
any angle ; and that in other substances the polarization
depends upon various circumstances which show how
idle and inapplicable is the account he thus gives of the
* Sec. 278.
374 PHILOSOPHY OF THE MECHANICO-CHEMICAL SCIENCES.
property. His self-complacent remark about the infer
ences of physics from experiment, is intended to recom
mend by comparison his own method of considering the
nature of things in themselves ; a mode of obtaining
physical truth which had been more than exhausted by
Aristotle, and out of which no new attempts have ex
tracted anything of value since his time.
14. Thus the general conclusion to which we are led
on this subject is, that the persuasion of the existence
and connexion or identity of various polarities in nature,
although very naturally admitted, and in many cases
interpreted and confirmed by observed facts, is of itself,
so far as we at present possess it, a very insecure guide
to scientific doctrines. When it is allowed to dictate
our theories, instead of animating and extending our
experimental researches, it leads only to errour, confusion,
obscurity, and mysticism.
This Fifth Book, on the subject of Polarities, is a
short one compared with most of the others. This
arises in a great measure from the circumstance that the
Idea of Polarity has only recently been apprehended and
applied, with any great degree of clearness, among phy
sical philosophers ; and is even yet probably entertained
in an obscure and ambiguous manner by most experi
mental inquirers. I have been desirous of not attempt
ing to bring forward any doctrines upon the subject,
except such as have been fully illustrated and exemplified
by the acknowledged progress of the physical sciences.
If I had been willing to discuss the various speculations
which have been published respecting the universal pre
valence of polarities in the universe, and their results in
every province of nature, I might easily have presented
this subject in a more extended form ; but this would
not have been consistent with my plan of tracing the
influence of scientific ideas only so far as they have really
OF THE CONNEXION OF POLARITIES. 375
aided in disclosing and developing scientific truths. And
as the influence of this idea is clearly distinguishable
both from those which precede and those which follow in
the character of the sciences to which it gives rise, and
appears likely to be hereafter of great extent and conse
quence, it seemed better to treat of it in a separate
Book, although of a brevity disproportioned to the
rest.
37G
BOOK VI.
THE PHILOSOPHY OF CHEMISTRY.
CHAPTER I.
ATTEMPTS TO CONCEIVE ELEMENTARY
COMPOSITION.
1. WE have now to bring into view, if possible, the
ideas and general principles which are involved in Che
mistry, — the science of the composition of bodies. For in
this as in other parts of human knowledge, we shall find
that there are certain ideas, deeply seated in the mind,
though shaped and unfolded by external observation,
which are necessary conditions of the existence of such
a science. These ideas it is, which impel man to such
a knowledge of the composition of bodies, which give
meaning to facts exhibiting this composition, and uni
versality to special truths discovered by experience.
These are the Ideas of Element and of Substance.
Unlike the idea of polarity, of which we treated in
the last Book, these ideas have been current in men's
minds from very early times, and formed the subject of
some of the first speculations of philosophers. It hap
pened however, as might have been expected, that in the
first attempts they were not clearly distinguished from
other notions, and were apprehended and applied in an
obscure and confused manner. We cannot better ex
hibit the peculiar character and meaning of these ideas
than by tracing the form which they have assumed and
CONCEPTION OF ELEMENTARY COMPOSITION. 377
the efficacy which they have exerted in these successive
essays. This, therefore, I shall endeavour to do, begin
ning with the Idea of Element.
2. That bodies are composed or made up of certain
parts, elements, or principles, is a conception which has
existed in men's minds from the beginning of the first
attempts at speculative knowledge. The doctrine of the
Four Elements, earth, air, fire and water, of which all
things in the universe were supposed to be constituted,
is one of the earliest forms in which this conception was
systematized ; and this doctrine is stated by various
authors to have existed as early as the times of the
ancient Egyptians"''. The words usually employed by
Greek writers to express these elements are dpxh a prin
ciple or beginning, and aToi-^elov, which probably meant
a letter (of a word) before it meant an element of a
compound. For the resolution of a word into its letters
is undoubtedly a remarkable instance of a successful
analysis performed at an early stage of man's history ;
and might very naturally supply a metaphor to denote
the analysis of substances into their intimate parts, when
men began to contemplate such an analysis as a subject
of speculation. The Latin word elementum itself, though
by its form it appears to be a derivative abstract term,
comes from some root now obsolete ; probably f from a
word signifying to grow or spring up.
The mode in which elements form the compound
bodies and determine their properties was at first, as
might be expected, vaguely and variously conceived. It
will, I trust, hereafter be made clear to the reader that
* Gilbert's Phys., L. i. c.iii.
t Vossius in voce. " Conjccto esse ab antiqua voco eleo pro oleo,
id est cresco : a qua significationc proles, suboles, adolescent : ut ab
juratum, juramentum ; ab adjitlum, adjnmcnhnn : sic ab delum,
elcmenhnn : quia intle oninia crcscunt ac nascuntur."
378 PHILOSOPHY OF C11EMIST11Y.
the relation of the elements to the compound involves a
peculiar and appropriate Fundamental Idea, not suscept
ible of being correctly represented by any comparison or
combination of other ideas, and guiding us to clear and
definite results only when it is illustrated and nourished
by an abundant supply of experimental facts. But at first
the peculiar and special notion which is required in a just
conception of the constitution of bodies was neither dis
cerned nor suspected ; and up to a very late period in the
history of chemistry, men went on attempting to appre
hend the constitution of bodies more clearly by substi
tuting for this obscure and recondite idea of Elementary
Composition, some other idea more obvious, more lumi
nous, and more familiar, such as the ideas of Resem
blance, Position, and mechanical Force. We shall briefly
speak of some of these attempts, and of the errours which
were thus introduced into speculations on the relations
of elements and compounds.
3. Compounds assumed to resemble their Elements. —
The first notion was that compounds derive their quali
ties from their elements by resemblance : — they are hot
in virtue of a hot element, heavy in virtue of a heavy
element, and so on. In this way the doctrine of the four
elements was framed ; for every body is either hot or
cold, moist or dry ; and by combining these qualities in
all possible ways, men devised four elementary sub
stances, as has been stated in the History"".
This assumption of the derivation of the qualities of
bodies from similar qualities in the elements was, as we
shall see, altogether baseless and unphilosophical, yet it
prevailed long and universally. It was the foundation of
medicine for a long period, both in Europe and Asia;
disorders being divided into hot, cold, and the like ; and
remedies being arranged according to similar distinctions.
* Hist. Ind Sci., B. i. c. ii. sect. 2.
CONCEPTION OF ELEMENTARY COMPOSITION. 379
Many readers will recollect, perhaps, the story* of the
indignation which the Persian physicians felt towards the
European, when he undertook to cure the ill effects of
cucumber upon the patient, by means of mercurial medi
cine : for cucumber, which is cold, could not be coun
teracted, they maintained, by mercury, which in their
classification is cold also. Similar views of the operation
of medicines might easily be traced in our own country.
A moment's reflection may convince us that when drugs
of any kind are subjected to the chemistry of the
human stomach and thus made to operate on the human
frame, it is utterly impossible to form the most remote
conjecture what the result will be from any such vague
notions of their qualities as the common use of our
senses can give. And in like manner the common ope
rations of chemistry give rise in almost every instance
to products which bear no resemblance to the materials
employed. The results of the furnace, the alembic, the
mixture, frequently have no visible likeness to the
ingredients operated upon. Iron becomes steel by the
addition of a little charcoal ; but what visible trace of
the charcoal is presented by the metal thus modified ?
The most beautiful colours are given to glass and
earthenware by minute portions of the ores of black or
dimjv metals, as iron and manganese. The worker in
o»
metal, the painter, the dyer, the vintner, the brewer,
all the artisans in short who deal with practical che
mistry, are able to teach the speculative chemist that
it is an utter mistake to expect that the qualities of the
elements shall be still discoverable, in an unaltered form,
in the compound. This first rude notion of an element,
that it determines the properties of bodies by resem
blance, must be utterly rejected and abandoned before
* See Hadji Baba.
380 PHILOSOPHY OF CHEMISTRY.
we can make any advance towards a true apprehension
of the constitution of bodies.
4. This step accordingly was made, when the hypo
thesis of the four elements was given up, and the doc
trine of the three Principles, Salt, Sulphur and Mercury,
was substituted in its place. For in making this change,
as I have remarked in the History*, the real advance
was the acknowledgment of the changes produced by
the chemist's operations as results to be accounted for
by the union and separation of substantial elements,
however great the changes, and however unlike the
product might be to the materials. And this step once
made, chemists wrent on constantly advancing towards
a truer view of the nature of an element, and conse
quently, towards a more satisfactory theory of chemical
operations.
5. Yet we may, I think, note one instance, even in
the works of eminent modern chemists, in which this
maxim, that we have no right to expect any resem
blance between the elements and the compound, is lost
sight of. I speak of certain classifications of mineral
substances. Berzelius, in his System of Mineral Arrange
ment, places sulphur next to the sulphur ets. But surely
this is an errour, involving the ancient assumption of
the resemblance of elements and compounds ; as if we
were to expect the sulphurets to bear a resemblance to
sulphur. All classifications are intended to bring toge
ther things resembling each other : the sulphurets of
metals have certain general resemblances to each other
which make them a tolerably distinct, well determined,
class of bodies. But sulphur has no resemblances with
these, and no analogies with them, either in physical
or even in chemical properties. It is a simple body;
* Hist, hid. Sci., B. iv. {.: i.
CONCEPTION OF ELEMENTARY COMPOSITION. 381
and both its resemblances and its analogies direct us to
place it along with other simple bodies, (selenium, and
phosphorus,) which, united with metals, produce com
pounds not very different from the sulphurets. Sulphur
cannot be, nor approach to being, a sulphuret ; we must
not confound what it is with what it makes. Sulphur
has its proper influence in determining the properties of
the compound into which it enters ; but it does not do
this according to resemblance of qualities, or according
to any principle which properly leads to propinquity in
classification. •
6. Compounds assumed to be determined by the Figure
of Elements. — I pass over the fanciful modes of represent
ing chemical changes which were employed by the Alche
mists ; for these strange inventions did little in leading
men towards a juster view of the relations of elements to
compounds. I proceed for an instant to the attempt to
substitute another obvious conception for the still obscure
notion of elementary composition. It was imagined that
all the properties of bodies and their mutual operations
might be accounted for by supposing them constituted of
particles of various forms, round or angular, pointed or
hooked, straight or spiral. This is a very ancient hypo
thesis, and a favourite one with many casual speculators
in all ages. Thus Lucretius undertakes to explain why
wine passes rapidly through a sieve and oil slowly, by
telling us that the latter substance has its particles either
larger than those of the other, or more hooked and inter
woven together. And he accounts for the difference of
sweet and bitter by supposing the particles in the former
case to be round and smooth, in the latter sharp and
jagged*. Similar assumptions prevailed in modern times
on the revival of the mechanical philosophy, and consti
tute a large part of the physical schemes of Descartes
* De Rrrttm Nalura, n. 390 siqq.
382 PHILOSOPHY OF CHEMISTRY.
and Gassendi. They were also adopted to a considerable
extent by the chemists. Acids were without hesitation
assumed to consist of sharp pointed particles ; which, " I
hope," Lemery says *, " no one will dispute, seeing every
one's experience does demonstrate it : he needs but taste
an acid to be satisfied of it, for it pricks the tongue like
anything keen and finely cut." Such an assumption is
not only altogether gratuitous and useless, but appears to
be founded in some degree upon a confusion in the meta
phorical and literal use of such words as keen and sharp.
The assumption once made, it was easy to accommodate
it, in a manner equally arbitrary, to other facts. "A
demonstrative and convincing proof that an acid does
consist of pointed parts is, that not only all acid salts do
crystallize into edges, but all dissolutions of different
things, caused by acid liquors, do assume this figure in
their crystallization. These crystals- consist of points
differing both in length and bigness one from another,
and this diversity must be attributed to the keener or
blunter edges of the different sorts of acids : and so like
wise this difference of the points in subtilty is the cause
that one acid can penetrate and dissolve with one sort of
mixt, that another can't rarify at all : Thus vinegar dis
solves lead, which aquafortis can't : aquafortis dissolves
quicksilver, which vinegar will not touch ; aqua regalis
dissolves gold, whenas aquafortis cannot meddle with it ;
on the contrary, aqua fortis dissolves silver, but can do
nothing with gold, and so of the rest."
The leading fact of the vehement combination and
complete union of acid and alkali readily suggested a fit
form for the particles of the latter class of substances.
" This effect," Lemery adds, " may make us reasonably
conjecture that an alkali is a terrestrious and solid mat
ter whose forms are figured after such a manner that the
* Chemistry, p. 25.
CONCEPTION OF ELEMENTARY COMPOSITION. 383
acid points entering in do strike and divide whatever
opposes their motion." And in a like spirit are the spe
culations in Dr. Mead's Mechanical Account of Poisons
(1745). Thus he explains the poisonous effect of corro
sive sublimate of mercury by saying* that the particles of
the salt are a kind of lamellae or blades to which the
mercury gives an additional weight. If resublimed with
three-fourths the quantity of mercury, it loses its corro-
siveness, (becoming calomel,) which arises from this, that
in sublimation " the crystalline blades are divided every
time more and more by the force of the fire ;" and " the
broken pieces of the crystals uniting into little masses of
differing figures from their former make, those cutting
points are now so much smaller that they cannot make
wounds deep enough to be equally mischievous and
deadly : and therefore do only vellicate and twitch the
sensible membranes of the stomach."
7. Among all this very fanciful and gratuitous assump
tion we may notice one true principle clearly introduced,
namely, that the suppositions which we make respecting
the forms of the elementary particles of bodies and their
mode of combination must be such as to explain the facts
of crystallization, as well as of mere chemical change.
This principle we shall hereafter have occasion to insist
upon further.
I now proceed to consider a more refined form of
assumption respecting the constitution of bodies, yet still
one in which a vain attempt is made to substitute for the
peculiar idea of chemical composition a more familiar
mechanical conception.
8. Compounds assumed to be determined by the Mecha
nical Attraction of the Elements. — When, in consequence
of the investigations and discoveries of Newton and his
predecessors, the conception of mechanical force had
* P. H>9.
384 PHILOSOPHY OF CHEMISTRY.
become clear and familiar, so far as the action of exter
nal forces upon a body was concerned, it was very natural
that the mathematicians who had pursued this train of
speculation should attempt to apply the same conception
to that mutual action of the internal parts of a body by
which they are held together. Newton himself had
pointed the way to this attempt, In the Preface to the
Principia, after speaking of what he has done in calcu
lating the effects of forces upon the planets, satellites,
&c., he adds, " Would it were permitted us to deduce the
other phenomena of nature from mechanical principles
by the same kind of reasoning. For many things move
me to suspect that all these phenomena depend upon
certain forces, by which the particles of bodies, through
causes not yet known, are either urged towards each
other, and cohere according to regular figures, or are
repelled and recede from each other ; which forces being
unknown, philosophers have hitherto made their attempts
upon nature in vain." The same thought is at a later
period followed out further in one of the Queries at the
end of the Opticks*. "Have not the small particles of
bodies certain Powers, Virtues, or Forces, by which they
act at a distance, not only upon the rays of light for
reflecting, refracting and inflecting them, but also upon
one another for producing a great part of the phenomena
of nature ?" And a little further on he proceeds to
apply this expressly to chemical changes. " When Salt
of Tartar runs per deliquium [or as we now express it,
deliquesces] is not this done by an attraction between
the particles of the Salt of Tartar and the particles of
the water which float in the air in the form of vapours ?
And why does not common salt, or saltpetre, or vitriol,
run per deliquium, but for want of such an attraction ? or
why does not Salt of Tartar draw more water out of the
" Query 31 .
CONCEPTION OF ELEMENTARY COMPOSITION. 583
air than in a certain proportion to its quantity, but for
want of an attractive force after it is saturated with
water?" He goes on to put a great number of similar
cases, all tending to the same point, that chemical com
binations cannot be conceived in any other way than as
an attraction of particles.
9. Succeeding speculators in his school attempted to
follow out this view. Dr. Frend, of Christ Church, in
1710, published his Prwlectiones Chymicce, in quibus
omnes fere Operationcs Chymicce ad xera PrincApia
ex ipsius Naturce Lcc/ilus rcdiyuntur. Oxonii liabitw.
This book is dedicated to Newton, and in the dedication,
the promise of advantage to chemistry from the influence
of the Newtonian discoveries is spoken of somewhat
largely, — much more largely, indeed, than has yet been
justified by the sequel. After declaring in strong terms
that the only prospect of improving science consists in
following the footsteps of Newton, the author adds,
" That force of attraction, of which you first so success
fully traced the influence in the heavenly bodies, ope
rates in the most minute corpuscles, as you long ago
hinted in your Principia, and have lately plainly shown
in your Opticks ; and this force we are only just begin
ning to perceive and to study. Under these circum
stances I have been desirous of trying what is the result
of this view in chemistry." The work opens formally
enough, with a statement of general mechanical prin
ciples, of which the most peculiar are these : — That
there exists an attractive force by which particles when
at very small distances from each other, are drawn to
gether; — that this force is different, according to the
different figure and density of the particles ; — that the
force may be greater on one side of a particle than on
the other; — that the force by which particles cohere
together arises from attraction, and is variously modi-
VOL. i. \v. P. C c
386 PHILOSOPHY OF CHEMISTRY.
fied according to the quantity of contacts." But these
principles are not applied in any definite manner to the
explanation of specific phenomena. He attempts, in
deed, the question of special solvents*. Why does aqua
fortis dissolve silver and not gold, while aqua regia
dissolves gold and not silver? which, he says, is the
most difficult question in chemistry, and which is cer
tainly a fundamental question in the formation of che
mical theory. He solves it by certain assumptions
respecting the forces of attraction of the particles, and
also the diameter of the particles of the acids and the
pores of the metals, all which suppositions are gratuitous.
10. We may observe further, that by speaking, as I
have stated that he does, of the figure of particles, he
mixes together the assumption of the last section with
the one which we are considering in this. This com
bination is very unphilosophical, or, to say the least,
very insufficient, since it makes a new hypothesis neces
sary. If a body be composed of cubical particles, held
together by their mutual attraction, by what force are
the parts of each cube held together ? In order to un
derstand their structure, we are obliged again to assume
a cohesive force of the second order, binding together
the particles of each particle. And therefore Newton
himself says f, very justly, "The parts of all homogeneal
hard bodies which fully touch each other, stick together
very strongly : and for explaning how this is, some have
invented hooked atoms, which is legging the question"
For (he means to imply,) how do the parts of the hook
stick together?
The same remark is applicable to all hypotheses in
which particles of a complex structure are assumed as
the constituents of bodies : for while we suppose bodies
and their known properties to result from the mutual
* P. 54. t Oplich; p. 304.
CONCEPTION OF ELEMENTARY COMPOSITION.
actions of these particles, we are compelled to suppose
the parts of each particle to be held together by forces
still more difficult to conceive, since they are disclosed
only by the properties of these particles, which as yet
are unknown. Yet Newton himself has not abstained
from such hypotheses : thus he says *, " A particle of a
salt may be compared to a chaos, being dense, hard, dry,
and earthy in the center, and moist and watery in the
circumference."
Since Newton's time the use of the term attraction,
as expressing the cause of the union of the chemical
elements of bodies, has been familiarly continued ; and
has, no doubt, been accompanied in the minds of many
persons with an obscure notion that chemical attraction
is, in some way, a kind of mechanical attraction of the
particles of bodies. Yet the doctrine that chemical " at
traction" and mechanical attraction are forces of the
same kind has never, so far as I am aware, been worked
out into a system of chemical theory ; nor even applied
with any distinctness as an explanation of any particular
chemical phenomena. Any such attenpt, indeed, could
only tend to bring more clearly into view the entire
inadequacy of such a mode of explanation. For the
leading phenomena of chemistry are all of such a nature
that no mechanical combination can serve to express
them, without an immense accumulation of additional
hypotheses. If we take as our problem the changes of
colour, transparency, texture, taste, odour, produced by
small changes in the ingredients, how can we expect to
give a mechanical account of these, till we can give
a mechanical account of colour, transparency, texture,
taste, odour, themselves ? And if our mechanical hypo
thesis of the elementary constitution of bodies does not
explain such phenomena as those changes, what can it
* Oplicks, p. 3G2.
CC2
388 miLosoriiY or CHEMISTRY.
explain, or what can be the value of it ? I do not here
insist upon a remark which will afterwards come before
us, that even crystalline form, a phenomenon of a far
more obviously mechanical nature than those just al
luded to, has never yet been in any degree explained by
such assumptions as this, that bodies consist of elemen
tary particles exerting forces of the same nature as the
central forces which we contemplate in Mechanics.
When therefore Newton asks, " When some stones,
as spar of lead, dissolved in proper menstruums, become
salts, do not these things show that salts are dry earth
and watery acid united by attraction f we may answer,
that this mode of expression appears to be intended to
identify chemical combination with mechanical attrac
tion ; — that there would be no objection to any such
identification, if we could, in that way, explain, or even
classify well, a collection of chemical facts ; but that
this has never yet been done by the help of such expres
sions. Till some advance of this kind can be pointed
out, we must necessarily consider the power which pro
duces chemical combination as a peculiar principle, a
special relation of the elements, not rightly expressed in
mechanical terms. And we now proceed to consider
this relation under the name by which it is most fami
liarly known.
CHAPTER II.
ESTABLISHMENT AND DEVELOPMENT OF THE
IDEA OF CHEMICAL AFFINITY.
1. THE earlier chemists did not commonly involve
themselves in the confusion into which the mechanical
philosophers ran, of comparing chemical to mechanical
forces. Their attention was engaged, and their ideas
IDEA OF CHEMICAL AFFINITY.
were moulded, by their own pursuits. They saw that
the connexion of elements and compounds with which
they had to deal, was a peculiar relation which must be
studied directly ; and which must be understood, if un
derstood at all, in itself, and not by comparison with a
different class of relations. At different periods of the
progress of chemistry, the conception of this relation,
still vague and obscure, was expressed in various man
ners; and at last this conception was clothed in tole
rably consistent phraseology, and the principles which it
involved were, by the united force of thought and expe
riment, brought into view.
2. The power by which the elements of bodies com
bine chemically, being, as we have seen, a peculiar agency,
different from mere mechanical connexion or attraction,
it is desirable to have it designated by a distinct and
peculiar name ; and the term Affinity has been employed
for that purpose by most modern chemists. The word
" affinity" in common language means, sometimes resem
blance, and sometimes relationship and ties of family.
It is from the latter sense that the metaphor is bor
rowed when we speak of " chemical affinity." By the
employment of this term we do not indicate resem
blance, but disposition to unite. Using the word in a
common unscientific manner, we might say that chlo
rine, bromine, and iodine, have a great natural affinity
with each other, for there are considerable resemblances
and analogies among them ; but these bodies have very
little chemical affinity for each other. The use of the
word in the former sense, of resemblance, can be traced
in earlier chemists; but it does not appear to have
acquired its peculiar chemical meaning till after Boer-
haave's time. Boerhaave, however, is the writer in
whom we first find a due apprehension of the peculiar
ity and importance of the Idea which it now expresses.
390 PHILOSOPHY OF CHEMISTRY.
When we make a chemical solution'"", he says, not only
are the particles of the dissolved body separated from
each other, but they are closely united to the particles
of the solvent. When aqua regia dissolves gold, do you
not see, he says to his hearers, that there must be be
tween each particle of the solvent and of the metal, a
mutual virtue by which each loves, unites with, and
holds the other (amat, unit, rctinet] ? The opinion pre
viously prevalent had been that the solvent merely
separates the parts of the body dissolved : and most
philosophers had conceived this separation as performed
by mechanical operations of the particles, resembling,
for instance, the operation of wedges breaking up a
block of timber. But Boerhaave forcibly and earnestly
points out the insufficiency of the conception. This, he
says, does not account for what we see. We have not
only a separation, but a new combination. There is a
force by which the particles of the solvent associate to
themselves the parts dissolved, not a force by which
they repel and dissever them. We are here to imagine
not mechanical action, not violent impulse, not antipathy,
but love, at least if love be the desire of uniting. (Xon
igitur hie etiam actiones mechanicse, non propulsiones
violenta?, non inimicitire cogitanda?, sed amicitisc, si amor
dicendus copulas cupido.) The novelty of this view is
evidenced by the mode in which he apologizes for intro
ducing it. " Fateor, paradoxa \\eec assertio." To Boer
haave, therefore, (especially considering his great influ
ence as a teacher of chemistry,) we may assign the
merit of first diffusing a proper view of Chemical Affinity
as a peculiar force, the origin of almost all chemical
changes and operations.
3. To Boerhaave is usually assigned also the credit
of introducing the norcl "affinity" among chemists; but
* Elemcnla Chemice. Lugd. Bat. 1732, p. 677-
IDEA OF CHEMICAL AFFINITY. 391
I do not find that the word is often used by him in this
sense; perhaps not at all*. But however this may be,
the term is, on many accounts well worthy to be pre
served, as I shall endeavour to show. Other terms were
used in the same sense during the early part of the
eighteenth century. Thus when Geoffrey, in 1718, laid
before the Academy of Paris his Tables of Affinities,
which perhaps did more- than any other event to fix the
Idea of Affinity, he termed them " Tables of the Rela
tions of Bodies ;" " Tables des Rapports :" speaking
however, also, of their " disposition to unite," and using
other phrases of the same import.
The term attraction, having been recommended by
Newton as a fit word to designate the force which pro
duces chemical combination, continued in great favour
in England, where the Newtonian philosophy was looked
upon as applicable to every branch of science. In
France, on the contrary, where Descartes still reigned
triumphant, " attraction," the watch-word of the enemy,
was a sound never uttered but with dislike and suspi
cion. In 1718 (in the notice of Geoffrey's Tables,) the
Secretary of the Academy, after pointing out some of
the peculiar circumstances of chemical combinations, says,
"Sympathies and attractions would suit well here, if
* See Dumas, Leqons de Phil. Chim.^ p. 3G4. Rees' Cyclopaedia^
Art. Chemistry. In the passage of Boerhaave to which I refer above,
affinitas is rather opposed to, than identified with, chemical combina
tion. "When, he says, the parts of the body to be dissolved are disse
vered by the solvent, why do they remain united to the particles of the
solvent, and why do not rather both the particles of the solvent and of
the dissolved body collect into homogeneous bodies by their affinity ?
"denuo so affinitate suae naturae colligant in corpora homogcnea ?" And
the answer is, because they possess another force which counteracts
this affinity of homogeneous particles, and makes compounds of dif
ferent elements. Affinity, in chemistry, now means the tendency of
different kinds of matter to unite : but it appears, as I have said, to
have acquired this sense since Boerhaave's time.
392 PHILOSOPHY OF CHEMISTRY.
there were such things." "Les sympathies, les attrac
tions conviendroient bien ici, si elles etaient quelque
chose." And at a later period, in 1731, having to write
the clone of Geoffroy after his death, he says, "He gave,
in 1718, a singular system, and a Table of Affinities, or
Relations of the different substances in chemistry. These
affinities gave uneasiness to some persons, who feared
that they were attractions in disguise, and all the more
dangerous in consequence of the seductive forms which
clever people have contrived to give them. It was found
in the sequel that this scruple might be got over."
This is the earliest published instance, so far as I am
aware, in which the word "affinity" is distinctly used
for the cause of chemical composition ; and taking into
account the circumstances, the word appears to have
been adopted in France in order to avoid the word
attraction, which had the taint of Newtonianism. Ac
cordingly we find the word affinite employed in the
works of French chemists from this time. Thus, in the
Transactions of the French Academy for 1746, in a
paper of Macquer's upon Arsenic, he says "% " On peut
facilement rendre raison de ces phenomenes par le moyen
des affinites que les differens substances qui entrant
dans ces combinaisons, out les uns avec les autres :" and
he proceeds to explain the facts by reference to Geof-
froy's Table. And in Macquer's Elements of Chemistry,
which appeared a few years later, the " affinity of com
position" is treated of as a leading part of the subject,
much in the same way as has been practised in such
books up to the present time. From this period, the
word appears to have become familiar to all European
chemists in the sense of which we are now speaking.
Thus, in the year 1758, the Academy of Sciences at
Rouen offered a prize for the best dissertation on Affinity.
* A. P. 1740, p. 201.
IDEA OF CHEMICAL AFFINITY.
The prize was shared between M. Limbourg of Theux,
near Liege, and M. Le Sage of Geneva*. About the
same time other persons (Manherrf, Nicolai J, and others)
wrote on the same subject, employing the same name.
Nevertheless, in 1775, the Swedish chemist Bergman,
pursuing still further this subject of Chemical Affinities,
and the expression of them by means of Tables, returned
again to the old Newtonian term; and designated the
disposition of a body to combine with one rather than
another of two others as elective attraction. And as his
work on Elective Attractions had great circulation and
great influence, this phrase has obtained a footing by the
side of Affinity, and both one and the other are now in
common use among chemists.
4. I have said above that the term Affinity is worthy
of being retained as a technical term. If we use the
word attraction in this case, we identify or compare
chemical with mechanical attraction ; from which iden
tification and comparison, as I have already remarked,
no one has yet been able to extract the means of ex
pressing any single scientific truth. If such an identi
fi 
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